Probability and Combinatorics

Probability Fundamentals helps us describe how likely something is to happen. Probability is used in games, weather forecasts, sports, and science. When we talk about probability, we are measuring chance.

A probability can be written as a number from 0 to 1.

• 0 means the event is impossible.
• 1 means the event is certain.
• A number between 0 and 1 means the event is possible, but not guaranteed.

For example, if the probability of rain is 0, it will not rain. If the probability is 1, it definitely will rain. If the probability is 0.5, rain is just as likely to happen as not happen.

Important idea: Probability compares the number of ways an event can happen to the total number of possible outcomes.

The basic probability formula is:

$$\text{Probability of an event} = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$$

We often shorten this to:

$$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

A favorable outcome is an outcome you want. The total outcomes are all the possible results.

1. Understanding the Probability Scale

It is helpful to picture probability on a scale from 0 to 1.

• 0 = impossible
• close to 0 = unlikely
• 0.5 = equally likely to happen or not happen
• close to 1 = likely
• 1 = certain

Here are some examples:

• Getting a 7 on a normal 6-sided die: probability 0
• Getting a number from 1 to 6 on a normal 6-sided die: probability 1
• Flipping heads on a fair coin: probability 0.5

Every probability must satisfy this rule:

$$0 \leq P(\text{event}) \leq 1$$

This means probability can never be less than 0 or greater than 1.

2. Writing Probability as a Fraction

Fractions are a common way to write probability. To find the probability, place the number of favorable outcomes over the total number of possible outcomes.

Suppose you spin a spinner with 4 equal sections: red, blue, green, and yellow. What is the probability of landing on blue?

There is 1 blue section out of 4 total sections, so:

$$P(\text{blue}) = \frac{1}{4}$$

This means the chance of landing on blue is one out of four.

If there are 3 red marbles in a bag of 10 marbles, then:

$$P(\text{red}) = \frac{3}{10}$$

Always make sure the denominator shows the total number of possible outcomes.

3. Writing Probability as a Decimal

A probability can also be written as a decimal. To change a fraction to a decimal, divide the numerator by the denominator.

For example:

$$\frac{1}{4} = 1 \div 4 = 0.25$$

So a probability of \(\frac{1}{4}\) is the same as 0.25.

Here are some common probability decimals:

• $$\frac{1}{2} = 0.5$$
• $$\frac{1}{4} = 0.25$$
• $$\frac{3}{4} = 0.75$$
• $$\frac{1}{10} = 0.1$$

Decimals are useful because they fit easily on the probability scale from 0 to 1.

4. Writing Probability as a Percent

A percent means “out of 100.” To change a decimal to a percent, multiply by 100.

For example, if the probability is 0.25:

$$0.25 \times 100 = 25\%$$

So these are all the same probability:

$$\frac{1}{4} = 0.25 = 25\%$$

Another example:

$$\frac{1}{2} = 0.5 = 50\%$$

Being able to switch between fractions, decimals, and percents is an important skill in probability.

5. Fair Outcomes

In many probability problems, we assume outcomes are equally likely. That means each outcome has the same chance of happening.

For example:

• A fair coin has 2 equally likely outcomes: heads and tails.
• A fair 6-sided die has 6 equally likely outcomes: 1, 2, 3, 4, 5, 6.

If outcomes are equally likely, the basic probability formula works directly.

If outcomes are not equally likely, then probability is more complicated. In this lesson, we focus on equally likely outcomes.

6. Complementary Probabilities

Sometimes it is easier to find the probability that an event does not happen. The probability of an event and the probability of its complement always add to 1.

$$P(\text{event}) + P(\text{not event}) = 1$$

For example, if the probability of drawing a red card is \(\frac{1}{4}\), then the probability of not drawing a red card is:

$$1 - \frac{1}{4} = \frac{3}{4}$$

This idea helps check your work. If your two probabilities do not add to 1, something is wrong.

Worked Example 1: Coin Flip

Question: A fair coin is flipped once. What is the probability of getting heads? Write the answer as a fraction, decimal, and percent.

Step 1: List the possible outcomes: heads, tails.

There are 2 total outcomes.

Step 2: Count the favorable outcomes.

There is 1 favorable outcome: heads.

Step 3: Write the probability as a fraction.

$$P(\text{heads}) = \frac{1}{2}$$

Step 4: Convert to a decimal.

$$\frac{1}{2} = 0.5$$

Step 5: Convert to a percent.

$$0.5 \times 100 = 50\%$$

Answer: $$\frac{1}{2}, \; 0.5, \; 50\%$$

Worked Example 2: Rolling a Die

Question: A fair 6-sided die is rolled once. What is the probability of rolling a number greater than 4?

Step 1: List the possible outcomes.

$$1, 2, 3, 4, 5, 6$$

There are 6 total outcomes.

Step 2: Find the favorable outcomes.

Numbers greater than 4 are 5 and 6.

There are 2 favorable outcomes.

Step 3: Write the probability.

$$P(\text{greater than 4}) = \frac{2}{6}$$

Step 4: Simplify the fraction.

$$\frac{2}{6} = \frac{1}{3}$$

Step 5: Convert if needed.

$$\frac{1}{3} \approx 0.333$$

$$0.333 \approx 33.3\%$$

Answer: $$\frac{1}{3}, \; 0.333\ldots, \; 33.3\% \text{ (about)}$$

Worked Example 3: Choosing from a Bag

Question: A bag contains 5 blue marbles, 3 green marbles, and 2 yellow marbles. One marble is chosen at random. What is the probability of choosing a green marble?

Step 1: Find the total number of marbles.

$$5 + 3 + 2 = 10$$

Step 2: Count the favorable outcomes.

There are 3 green marbles.

Step 3: Write the probability.

$$P(\text{green}) = \frac{3}{10}$$

Step 4: Convert to decimal and percent.

$$\frac{3}{10} = 0.3$$

$$0.3 \times 100 = 30\%$$

Answer: $$\frac{3}{10}, \; 0.3, \; 30\%$$

Worked Example 4: Using the Complement

Question: A spinner has 8 equal sections. 3 sections are red and 5 sections are not red. What is the probability of not landing on red?

Method 1: Count directly

There are 5 sections that are not red out of 8 total sections.

$$P(\text{not red}) = \frac{5}{8}$$

Method 2: Use the complement

First find the probability of red:

$$P(\text{red}) = \frac{3}{8}$$

Then subtract from 1:

$$P(\text{not red}) = 1 - \frac{3}{8} = \frac{5}{8}$$

Convert to decimal and percent:

$$\frac{5}{8} = 0.625 = 62.5\%$$

Answer: $$\frac{5}{8}, \; 0.625, \; 62.5\%$$

7. Tips for Solving Probability Problems

• Read carefully. Decide what event you are finding the probability of.
• Count all possible outcomes. This is the denominator.
• Count favorable outcomes. This is the numerator.
• Simplify the fraction if possible.
• Convert to decimal or percent if asked.
• Check your answer. It must be between 0 and 1, or between 0% and 100%.

8. Common Mistakes to Avoid

• Forgetting the total number of outcomes. Make sure the denominator includes every possible result.
• Using numbers outside the probability range. Probabilities cannot be less than 0 or greater than 1.
• Mixing up favorable and total outcomes. The event you want goes on top.
• Not simplifying the fraction. For example, \(\frac{2}{6}\) should be simplified to \(\frac{1}{3}\).
• Incorrect percent conversion. Multiply the decimal by 100, not by 10.

9. Quick Practice Ideas

Try these on your own:

1. A fair die is rolled. What is the probability of rolling an even number?
2. A bag has 7 red pencils and 3 blue pencils. What is the probability of choosing a blue pencil?
3. A spinner has 5 equal sections. What is the probability of not landing on one chosen section?

Check that your answers can be written as a fraction, decimal, and percent.

Summary

Probability tells how likely an event is to happen. It is always a number from 0 to 1, where 0 means impossible and 1 means certain.

To find probability, use:

$$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

You should also be able to write probabilities as fractions, decimals, and percents. For example, \(\frac{1}{2} = 0.5 = 50\%\).

When solving problems, count carefully, keep probabilities between 0 and 1, and use the complement rule when it helps.

Lesson: Theoretical vs. Experimental Probability

Probability helps us describe how likely something is to happen. In everyday life, we use probability when we talk about weather forecasts, games, sports, and chance events like rolling dice or picking a card.

In this lesson, you will learn the difference between theoretical probability and experimental probability. These are two ways of thinking about chance. One is based on math and what should happen. The other is based on actual results and what did happen.

Understanding both is important because real-life results do not always match the exact probability we expect. But over many trials, experimental results often get closer to theoretical results.

1. What is probability?

Probability is a number that tells how likely an event is to happen. It can be written as a fraction, decimal, or percent.

• A probability of 0 means the event is impossible.
• A probability of 1 means the event is certain.
• Most probabilities are between 0 and 1.

For example, when flipping a fair coin, the probability of landing on heads is \(\frac{1}{2}\), or 0.5, or 50%.

2. What is theoretical probability?

Theoretical probability is based on what we know about all the possible outcomes, assuming the situation is fair.

We use this formula:

A favorable outcome is an outcome you want. The total number of possible outcomes means all the outcomes that could happen.

For example, when rolling a fair 6-sided die, the possible outcomes are 1, 2, 3, 4, 5, and 6.

If you want to roll a 4, there is 1 favorable outcome out of 6 possible outcomes.

If you want to roll an even number, the favorable outcomes are 2, 4, and 6. That is 3 favorable outcomes out of 6 possible outcomes.

3. What is experimental probability?

Experimental probability is based on actual results from an experiment or trial.

We use this formula:

A trial is one time you do the experiment, like one coin flip or one die roll.

Suppose you flip a coin 10 times and get 7 heads. Then the experimental probability of heads is

This does not match the theoretical probability of \(\frac{1}{2}\), but that is okay. Experimental results can be different from expected results, especially when the number of trials is small.

4. The key difference

• Theoretical probability is what should happen based on math.
• Experimental probability is what actually happens in real trials.

Think of it this way:

• Theoretical probability uses the outcome space.
• Experimental probability uses data collected from an experiment.

5. Why are they sometimes different?

Even if a game or experiment is fair, results do not always come out exactly as expected in a small number of tries.

For example, if a coin is fair, heads and tails are equally likely. But if you flip it only 4 times, you might get 3 heads and 1 tail. That does not mean the coin is unfair. It just means chance can vary.

When the number of trials gets larger, the experimental probability usually gets closer to the theoretical probability.

So if you flip a coin 10 times, your results may be far from \(\frac{1}{2}\). But if you flip it 1,000 times, the fraction of heads will often be much closer to \(\frac{1}{2}\).

6. Worked Example 1: Theoretical probability with a spinner

A spinner has 8 equal sections. 3 sections are red, 2 are blue, 2 are green, and 1 is yellow.

Question: What is the theoretical probability of landing on blue?

Step 1: Count the favorable outcomes.

There are 2 blue sections.

Step 2: Count the total possible outcomes.

There are 8 sections total.

Step 3: Write the probability.

Answer: The theoretical probability of landing on blue is \(\frac{1}{4}\).

7. Worked Example 2: Experimental probability with a spinner

The same spinner is spun 20 times. It lands on blue 7 times.

Question: What is the experimental probability of landing on blue?

Step 1: Count how many times the event happened.

The spinner landed on blue 7 times.

Step 2: Count the total number of trials.

There were 20 spins.

Step 3: Write the probability.

Answer: The experimental probability of landing on blue is \(\frac{7}{20}\).

Compare: The theoretical probability is \(\frac{1}{4} = \frac{5}{20}\), but the experimental probability is \(\frac{7}{20}\). They are close, but not the same.

8. Worked Example 3: Comparing theoretical and experimental probability

A number cube with faces 1 through 6 is rolled 30 times. The results show that a 6 was rolled 8 times.

Question 1: What is the theoretical probability of rolling a 6?

There is 1 favorable outcome (rolling a 6) out of 6 total possible outcomes.

Question 2: What is the experimental probability of rolling a 6?

A 6 happened 8 times out of 30 rolls.

Question 3: Which probability is larger?

Let us compare:

The experimental probability is larger.

Question 4: Does this mean the die is unfair?

Not necessarily. In only 30 rolls, results can vary. To decide whether a die might be unfair, you would need many more trials.

9. Worked Example 4: Compound event with two coins

A compound event is an event with more than one simple action, like flipping two coins.

Suppose you flip two fair coins.

Question: What is the theoretical probability of getting exactly one head?

Step 1: List the possible outcomes.

• HH
• HT
• TH
• TT

There are 4 total possible outcomes.

Step 2: Find the favorable outcomes.

Exactly one head happens in:

• HT
• TH

So there are 2 favorable outcomes.

Step 3: Write the theoretical probability.

Now suppose two coins are flipped 12 times, and exactly one head happens 5 times.

The experimental probability is

Compare:

• Theoretical probability: \(\frac{1}{2}\)
• Experimental probability: \(\frac{5}{12}\)

These are close, but not equal.

10. How to solve probability questions

When you are asked for theoretical probability, follow these steps:

1. List or think about all possible outcomes.
2. Count how many outcomes fit the event.
3. Use the formula \(\frac{\text{favorable}}{\text{total}}\).

When you are asked for experimental probability, follow these steps:

1. Look at the data or results.
2. Count how many times the event happened.
3. Count the total number of trials.
4. Use the formula \(\frac{\text{event happened}}{\text{total trials}}\).

11. Important things to watch for

• Do not mix up possible outcomes with actual outcomes. Possible outcomes are for theoretical probability. Actual outcomes are for experimental probability.
• Make sure the situation is fair when finding theoretical probability. If a spinner has equal sections, each section is equally likely.
• Use the total number of trials for experimental probability, not the total number of possible outcomes.
• Simplify fractions when needed.

12. Quick comparison chart

• Theoretical probability: Based on math, uses possible outcomes, predicts what should happen.
• Experimental probability: Based on data, uses actual results, shows what did happen.

13. Practice thinking

Ask yourself these questions when solving:

• Am I using the outcomes that could happen, or the outcomes that did happen?
• Is this asking for a prediction or for a result from an experiment?
• What is the numerator?
• What is the denominator?

14. Summary

Theoretical probability tells the chance of an event based on all possible outcomes in a fair situation.

$$\text{Theoretical Probability} = \frac{\text{favorable outcomes}}{\text{total possible outcomes}}$$

Experimental probability tells the chance of an event based on actual trial results.

$$\text{Experimental Probability} = \frac{\text{times the event happened}}{\text{total number of trials}}$$

Theoretical probability is what we expect from math. Experimental probability is what we observe in real life. They may be different, especially with a small number of trials, but with more trials, they often become closer.

Law of Large Numbers is a big idea in probability that helps us understand what happens when we repeat a random experiment many times.

It tells us that when we do more and more trials, the experimental probability usually gets closer and closer to the theoretical probability.

In simple words: small numbers of trials can be noisy and surprising, but large numbers of trials tend to show the expected pattern.

Important words

• Probability: how likely something is to happen.
• Theoretical probability: the probability we expect using math before doing the experiment.
• Experimental probability: the probability we get from actual results.
• Trial: one repeat of an experiment, like one coin flip or one die roll.

The formula for experimental probability is:

$$\text{Experimental probability} = \frac{\text{number of times the event happens}}{\text{total number of trials}}$$

The formula for theoretical probability is:

$$\text{Theoretical probability} = \frac{\text{number of favorable outcomes}}{\text{number of total possible outcomes}}$$

For example, when flipping a fair coin, there are 2 possible outcomes: heads or tails.

The theoretical probability of getting heads is:

$$P(\text{heads}) = \frac{1}{2}$$

That means we expect heads about half the time. But that does not mean every 2 flips must give exactly 1 head and 1 tail.

This is where the Law of Large Numbers matters. In a small number of flips, results can be uneven. In a large number of flips, the results usually become more balanced.

Why does this happen?

Random events can bounce around a lot at first. If you only do a few trials, one outcome might happen much more than expected just by chance.

But as you collect more data, the effect of those early surprising results becomes smaller. The overall proportion starts settling near the theoretical probability.

This does not mean the results become perfect. It means they usually become closer to the expected probability.

A helpful way to think about it

Imagine flipping a coin 4 times and getting 3 heads. The experimental probability of heads is:

$$\frac{3}{4} = 0.75$$

That is quite different from the theoretical probability of \(0.5\).

Now imagine flipping the coin 400 times and getting 204 heads. The experimental probability is:

$$\frac{204}{400} = 0.51$$

That is much closer to \(0.5\).

This is exactly what the Law of Large Numbers describes.

Worked Example 1: Coin flips

A student flips a fair coin 10 times and gets 7 heads.

Find the experimental probability of heads, and compare it to the theoretical probability.

Step 1: Find the experimental probability

$$\text{Experimental probability of heads} = \frac{7}{10} = 0.7$$

Step 2: Find the theoretical probability

For a fair coin:

$$P(\text{heads}) = \frac{1}{2} = 0.5$$

Step 3: Compare

The experimental probability, \(0.7\), is higher than the theoretical probability, \(0.5\).

This does not mean the coin is unfair. With only 10 trials, results can vary a lot.

What the Law of Large Numbers says: if the student keeps flipping the coin many more times, the experimental probability will likely move closer to \(0.5\).

Worked Example 2: Rolling a die

A fair 6-sided die is rolled 60 times. The number 4 appears 8 times.

Find the experimental probability of rolling a 4, and compare it to the theoretical probability.

Step 1: Experimental probability

$$\text{Experimental probability of rolling a 4} = \frac{8}{60} = \frac{2}{15} \approx 0.133$$

Step 2: Theoretical probability

A fair die has 6 equally likely outcomes, and only one of them is 4.

$$P(4) = \frac{1}{6} \approx 0.167$$

Step 3: Compare

The experimental probability \(0.133\) is a little less than the theoretical probability \(0.167\).

Since 60 rolls is a moderate number of trials, the experimental result may be somewhat close, but not exact.

If we rolled the die hundreds or thousands of times, the experimental probability would usually get even closer to \(\frac{1}{6}\).

Worked Example 3: Colored marbles simulation

A bag has 3 red marbles and 1 blue marble. A marble is chosen, the color is recorded, and then the marble is put back. This is repeated 40 times. Red is chosen 32 times.

Find the theoretical and experimental probabilities of choosing red.

Step 1: Theoretical probability

There are 4 marbles total, and 3 are red.

$$P(\text{red}) = \frac{3}{4} = 0.75$$

Step 2: Experimental probability

$$\text{Experimental probability of red} = \frac{32}{40} = \frac{4}{5} = 0.8$$

Step 3: Compare

The experimental probability is \(0.8\), and the theoretical probability is \(0.75\).

These values are close. That makes sense because 40 trials is enough to begin showing the expected pattern.

If the experiment were repeated 400 times instead of 40, we would expect the experimental probability to likely be even closer to \(0.75\).

Worked Example 4: Looking at how probability changes with more trials

A spinner has 4 equal sections: 1 green, 1 yellow, 1 blue, and 1 red.

The theoretical probability of landing on green is:

$$P(\text{green}) = \frac{1}{4} = 0.25$$

A class records these results:

• After 8 spins, green appeared 4 times.
• After 40 spins, green appeared 12 times.
• After 200 spins, green appeared 51 times.

Let us find the experimental probability each time.

After 8 spins:

$$\frac{4}{8} = 0.5$$

After 40 spins:

$$\frac{12}{40} = 0.3$$

After 200 spins:

$$\frac{51}{200} = 0.255$$

Now compare each result to the theoretical probability \(0.25\).

• After 8 spins: \(0.5\) is far from \(0.25\).
• After 40 spins: \(0.3\) is closer.
• After 200 spins: \(0.255\) is very close.

This shows the Law of Large Numbers clearly: as the number of trials increases, the experimental probability gets closer to the theoretical probability.

What the Law of Large Numbers does NOT mean

• It does not mean every short experiment will match the expected probability.
• It does not mean outcomes must “take turns” fairly in the short run.
• It does not mean a result is “due” just because it has not happened recently.

For example, if a coin lands on heads 5 times in a row, that does not mean tails is guaranteed on the next flip. The next flip is still just one flip, and for a fair coin:

$$P(\text{tails}) = \frac{1}{2}$$

The Law of Large Numbers is about the overall pattern across many trials, not about changing what happens on the very next trial.

How to recognize this idea in a problem

You may be using the Law of Large Numbers if a question asks you to:

• compare experimental probability and theoretical probability,
• decide what happens when the number of trials increases,
• look at results from a simulation,
• explain why a larger sample gives a more reliable estimate.

Steps for solving problems

1. Identify the event, such as rolling a 6 or drawing a red marble.
2. Find the theoretical probability using the possible outcomes.
3. Find the experimental probability using the data collected.
4. Compare the two probabilities.
5. If the number of trials increases, predict that the experimental probability will likely move closer to the theoretical probability.

Quick check

If a fair coin is flipped 20 times and lands on heads 14 times, the experimental probability of heads is:

$$\frac{14}{20} = 0.7$$

The theoretical probability is:

$$\frac{1}{2} = 0.5$$

If the coin is flipped 2,000 times instead, would you expect the experimental probability to be closer to \(0.5\) or farther away?

Answer: closer to \(0.5\), because of the Law of Large Numbers.

Brief Summary

The Law of Large Numbers says that when a random experiment is repeated many times, the experimental probability tends to get closer to the theoretical probability.

Small numbers of trials can give surprising results, but large numbers of trials usually show the expected pattern more clearly.

So, if you want a probability experiment to better match what math predicts, do more trials.

Sample Space and Tree Diagrams

When we do an activity that has different possible results, those results are called outcomes. In probability, it is very important to know all possible outcomes before answering questions.

The complete list of all possible outcomes is called the sample space.

For example, if you flip a coin once, the sample space is:

$$\{\text{Heads},\text{Tails}\}$$

If you roll a number cube once, the sample space is:

$$\{1,2,3,4,5,6\}$$

Knowing the sample space helps us count outcomes carefully and find probabilities correctly.

Why sample spaces matter

If you miss an outcome, your answer will be wrong. If you list an outcome twice, your answer will also be wrong. That is why we need a systematic way to organize outcomes.

Some useful ways to show a sample space are:

• A list of outcomes
• A table or matrix of outcomes
• A tree diagram for events that happen in steps

What is a tree diagram?

A tree diagram is a branching picture that shows the possible outcomes of a multi-stage event. Each branch shows one choice or result at each stage.

Tree diagrams are especially helpful when:

• there is more than one action, like flipping a coin and then rolling a number cube,
• the order matters, or
• you want to make sure you do not forget any outcomes.

How to make a tree diagram

1. Write the possible outcomes for the first stage.
2. From each of those outcomes, draw branches for the second stage.
3. If there is a third stage, keep branching.
4. At the end of each path, write the full outcome.
5. Count all the final outcomes to get the sample space.

Worked Example 1: Flipping a coin twice

Suppose you flip a coin two times. Each flip can be Heads \,\((H)\) or Tails \,\((T)\).

First flip: \(H\) or \(T\)

Second flip: \(H\) or \(T\)

Using a tree idea:

• Start with \(H\), then branch to \(H\) and \(T\)
• Start with \(T\), then branch to \(H\) and \(T\)

The sample space is:

$$\{HH, HT, TH, TT\}$$

There are 4 possible outcomes.

Notice that \(HT\) and \(TH\) are different because the order of the flips is different.

Worked Example 2: Rolling a number cube and flipping a coin

Now suppose you roll a number cube and then flip a coin.

The number cube outcomes are:

$$\{1,2,3,4,5,6\}$$

The coin outcomes are:

$$\{H,T\}$$

To make the sample space, pair each number with each coin result:

$$\{1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T\}$$

There are:

$$6 \times 2 = 12$$

So the sample space has 12 outcomes.

This could also be shown in a matrix with 6 rows and 2 columns, or with a tree diagram that starts with the 6 number outcomes and then branches to \(H\) and \(T\).

Worked Example 3: Choosing an outfit

A student has:

• 2 shirts: red and blue
• 3 pairs of pants: black, gray, and tan

How many different outfits are possible?

This is a two-stage event:

• Choose a shirt
• Choose pants

Make branches from each shirt to each pair of pants.

The sample space is:

$$\{\text{red-black},\text{red-gray},\text{red-tan},\text{blue-black},\text{blue-gray},\text{blue-tan}\}$$

There are:

$$2 \times 3 = 6$$

So there are 6 possible outfits.

This shows that tree diagrams are not only for coins and dice. They can help with any situation that happens in steps.

Worked Example 4: Two spins

A spinner has 3 equal sections labeled \(A\), \(B\), and \(C\). It is spun twice.

First spin outcomes:

$$\{A,B,C\}$$

Second spin outcomes:

$$\{A,B,C\}$$

Use a tree diagram or organized list:

$$\{AA, AB, AC, BA, BB, BC, CA, CB, CC\}$$

There are:

$$3 \times 3 = 9$$

So the sample space has 9 outcomes.

If you wanted the outcomes where both spins match, they would be:

$$\{AA, BB, CC\}$$

That is 3 outcomes out of 9 total.

Using multiplication to count outcomes

When an event happens in stages, you can often multiply the number of choices in each stage.

For example:

• 2 coin outcomes and 6 number cube outcomes gives \(2 \times 6 = 12\)
• 3 spinner outcomes and 3 spinner outcomes gives \(3 \times 3 = 9\)
• 2 shirts and 3 pants gives \(2 \times 3 = 6\)

This works when each choice in one stage can be matched with every choice in the next stage.

Lists, matrices, and tree diagrams

All three methods can show the same sample space. The best method depends on the problem.

• List: Good when there are only a few outcomes.
• Matrix or table: Good when two sets of outcomes are being paired.
• Tree diagram: Good when there are several steps or when order matters.

For example, if you flip a coin and roll a number cube, a table works well. If you flip a coin three times, a tree diagram is often easier to follow.

Common mistakes to avoid

• Forgetting outcomes: Make sure every branch is completed.
• Repeating outcomes: Do not count the same outcome more than once.
• Ignoring order: In some experiments, order matters. For example, \(HT\) is not the same as \(TH\).
• Stopping too early: If there are three stages, your tree needs three levels of branches.

How to check your work

After making a sample space, ask yourself:

• Did I include every possible choice at each stage?
• Did I write each final outcome clearly?
• Did I count each outcome only once?
• Can I check the total by multiplication?

Quick Practice

1. A coin is flipped and a spinner with colors red and green is spun. The sample space is:

$$\{H\text{red}, H\text{green}, T\text{red}, T\text{green}\}$$

2. A sandwich shop offers 2 breads and 4 fillings. The number of different sandwiches is:

$$2 \times 4 = 8$$

3. A number cube is rolled twice. The total number of outcomes is:

$$6 \times 6 = 36$$

You do not always need to list all 36 outcomes if the question only asks for the total number, but a tree diagram or organized table can help if you need the full sample space.

Summary

A sample space is the complete set of all possible outcomes. When an event happens in more than one step, a tree diagram helps organize the outcomes so none are missed.

You can also use lists or matrices to show the same information. To find the total number of outcomes, multiply the number of choices at each stage when every choice can pair with every other choice.

Fundamental Counting Principle is a quick way to count how many different outcomes are possible in a situation with steps.

If one choice is followed by another choice, and then maybe another, you can find the total number of outcomes by multiplying the number of options at each step.

This idea is very useful in probability and combinatorics because it helps us find the size of a sample space, which is the full set of all possible outcomes.

For example, if you choose 1 shirt and 1 pair of pants, and there are 3 shirts and 2 pairs of pants, then the total number of outfits is:

$$3 \times 2 = 6$$

This works because for each shirt, there are 2 possible pairs of pants.

When to Use the Fundamental Counting Principle

Use it when:

• There are separate steps in a process.
• Each step has a certain number of choices.
• You want the total number of possible outcomes.

You can think of it like this:

$$\text{Total outcomes} = (\text{choices for step 1}) \times (\text{choices for step 2}) \times (\text{choices for step 3}) \times \cdots$$

If there are 4 ways to do the first step and 5 ways to do the second step, then:

$$4 \times 5 = 20$$

So there are 20 total outcomes.

Why Multiplication Works

Multiplication works because every option in one step can be paired with every option in the next step.

Suppose you have 2 drinks and 3 snacks.

• Drink 1 can go with 3 snacks.
• Drink 2 can also go with 3 snacks.

That gives:

$$2 \times 3 = 6$$

possible snack-and-drink combinations.

You do not add here, because you are not choosing just one category. You are making one choice from each category.

Worked Example 1: Ice Cream Choices

A shop has 4 flavors of ice cream and 3 types of cones. How many different ice cream orders are possible if you choose 1 flavor and 1 cone?

Step 1: Count the choices for flavor: 4

Step 2: Count the choices for cone: 3

Step 3: Multiply:

$$4 \times 3 = 12$$

Answer: There are 12 different ice cream orders.

Worked Example 2: Creating a Meal

A lunch special lets you choose 1 sandwich, 1 drink, and 1 fruit.

• 5 sandwich choices
• 2 drink choices
• 4 fruit choices

How many different lunches can be made?

There are 3 steps, so multiply all the choices:

$$5 \times 2 \times 4 = 40$$

Answer: There are 40 possible lunches.

Notice that adding would not make sense here. You are not picking only one item total. You are picking one item from each group.

Worked Example 3: Rolling Number Cubes

You roll two number cubes. Each number cube has 6 possible outcomes.

Step 1: First cube: 6 choices

Step 2: Second cube: 6 choices

Multiply:

$$6 \times 6 = 36$$

Answer: There are 36 possible outcomes.

This means the sample space for rolling two number cubes has 36 outcomes.

For example, 1,1, 1,2, and 2,1 are different outcomes because order matters: the first cube and second cube are separate steps.

Worked Example 4: Making a Code

A locker code uses 1 letter followed by 1 digit.

• There are 26 possible letters.
• There are 10 possible digits, from 0 to 9.

How many different codes are possible?

Multiply the choices for each step:

$$26 \times 10 = 260$$

Answer: There are 260 different codes.

If the code had 2 letters followed by 1 digit, then the number of outcomes would be:

$$26 \times 26 \times 10 = 6760$$

Each new step means another multiplication.

How This Connects to Probability

In probability, we often want to know how many outcomes are possible altogether.

The Fundamental Counting Principle helps us find that total quickly. Once we know the total number of outcomes in the sample space, we can compare it to the number of outcomes we want.

For example, when rolling two number cubes, there are 36 total outcomes. If you wanted the probability of getting a sum of 7, you would first know the sample space has size 36 because:

$$6 \times 6 = 36$$

So the counting principle helps build the foundation for probability problems.

Common Mistakes to Avoid

• Adding instead of multiplying: If the problem has steps and you choose from each step, multiply.
• Forgetting a step: Make sure every choice category is included.
• Mixing up order: Sometimes order matters because the steps are different, like first roll and second roll.
• Using the wrong number of choices: Count carefully before multiplying.

Helpful Strategy

1. Identify each step in the event.
2. Count how many choices there are for each step.
3. Multiply the numbers.
4. Check whether your answer makes sense.

You can organize your thinking with a simple model like this:

$$\text{Total} = a \times b \times c$$

where \(a\), \(b\), and \(c\) are the numbers of choices at each step.

Quick Practice Ideas

Try asking yourself these questions:

• If there are 3 kinds of cereal and 2 kinds of milk, how many breakfast combinations are there?
• If a flag uses 1 of 4 colors for the top stripe and 1 of 5 colors for the bottom stripe, how many flags are possible?
• If you flip a coin and then spin a spinner with 4 sections, how many outcomes are possible?

For those, you would multiply the choices at each step.

Summary

The Fundamental Counting Principle says that if an event happens in steps, you can find the total number of outcomes by multiplying the number of choices in each step.

This helps you count combinations quickly and find the size of a sample space.

Remember: separate steps mean multiply.

Comparing Experimental and Theoretical Probability

Probability helps us describe how likely something is to happen. For example, if you flip a coin, roll a number cube, or pick a colored marble from a bag, probability tells us the chance of each outcome.

There are two important kinds of probability we use in math: theoretical probability and experimental probability. In this lesson, you will learn what each one means, how to calculate them, and how to compare them.

You will also learn an important idea called the Law of Large Numbers. This law says that when we repeat an experiment many times, the experimental probability usually gets closer and closer to the theoretical probability.

1. Theoretical Probability

Theoretical probability is what we expect to happen based on reasoning, without actually doing the experiment.

We use this formula:

$$\text{Theoretical Probability} = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$$

A favorable outcome is an outcome we want. The total number of possible outcomes is all the outcomes that could happen.

Example: If you roll a fair 6-sided number cube, what is the theoretical probability of rolling a 3?

There is 1 favorable outcome: rolling a 3.

There are 6 possible outcomes: 1, 2, 3, 4, 5, 6.

So,

$$P(3)=\frac{1}{6}$$

This means the chance of rolling a 3 is 1 out of 6.

2. Experimental Probability

Experimental probability is based on what actually happens when we do an experiment.

We use this formula:

$$\text{Experimental Probability} = \frac{\text{number of times the event happens}}{\text{total number of trials}}$$

A trial is one time you do the experiment. If you flip a coin 20 times, that is 20 trials.

Example: Suppose you roll a number cube 12 times and get a 3 only 4 times.

Then the experimental probability of rolling a 3 is

$$\frac{4}{12}=\frac{1}{3}$$

Notice that \(\frac{1}{3}\) is not the same as the theoretical probability \(\frac{1}{6}\). That is okay. In a small number of trials, results may be different from what we expect.

3. Comparing Theoretical and Experimental Probability

To compare them, ask:

• What should happen in theory?
• What actually happened in the experiment?
• How close are the two probabilities?

Sometimes the two probabilities are very close. Sometimes they are not. A small experiment can have surprising results just by chance.

4. The Law of Large Numbers

The Law of Large Numbers says that as the number of trials gets larger, the experimental probability usually gets closer to the theoretical probability.

This does not mean the results will become exactly the same every time. It means that with more and more trials, the results tend to even out and better match what probability predicts.

Example idea: If you flip a fair coin only 4 times, you might get 3 heads and 1 tail. That gives an experimental probability of heads of

$$\frac{3}{4}$$

But the theoretical probability of heads is

$$\frac{1}{2}$$

Those are not very close.

If you flip the same coin 200 times, the number of heads may be much closer to half of 200, which is 100. The experimental probability may be something like

$$\frac{98}{200}=0.49$$

That is very close to the theoretical probability of \(0.5\).

Why does this happen?

In just a few trials, chance can make results look uneven. But over many trials, the extra heads and extra tails begin to balance out more.

Worked Example 1: Coin Flip

A fair coin is flipped 10 times. It lands on heads 6 times.

Step 1: Find the theoretical probability of heads.

A fair coin has 2 possible outcomes: heads or tails.

There is 1 favorable outcome for heads.

$$P(\text{heads})=\frac{1}{2}$$

Step 2: Find the experimental probability of heads.

Heads happened 6 times out of 10 trials.

$$\frac{6}{10}=\frac{3}{5}$$

Step 3: Compare them.

• Theoretical probability: \(\frac{1}{2}=0.5\)
• Experimental probability: \(\frac{3}{5}=0.6\)

The experimental probability is a little higher than the theoretical probability. Since there were only 10 flips, this difference is not surprising.

Worked Example 2: Rolling a Number Cube

A fair 6-sided number cube is rolled 30 times. The number 5 appears 7 times.

Step 1: Find the theoretical probability of rolling a 5.

There is 1 favorable outcome and 6 total outcomes.

$$P(5)=\frac{1}{6}$$

Step 2: Find the experimental probability.

$$\frac{7}{30}$$

Step 3: Compare them.

The theoretical probability is

$$\frac{1}{6}\approx 0.167$$

The experimental probability is

$$\frac{7}{30}\approx 0.233$$

These are not exactly the same, but they are reasonably close. If the cube were rolled many more times, the experimental probability might move closer to \(\frac{1}{6}\).

Worked Example 3: Colored Marbles

A bag has 3 red marbles and 1 blue marble. A marble is chosen, the color is recorded, and then the marble is put back. This is repeated 40 times. Red is chosen 28 times.

Step 1: Find the theoretical probability of choosing red.

There are 3 red marbles out of 4 total marbles.

$$P(\text{red})=\frac{3}{4}$$

Step 2: Find the experimental probability of choosing red.

$$\frac{28}{40}=\frac{7}{10}$$

Step 3: Compare them.

• Theoretical probability: \(\frac{3}{4}=0.75\)
• Experimental probability: \(\frac{7}{10}=0.7\)

The experimental probability is close to the theoretical probability, but it is a little lower.

Step 4: Think about the Law of Large Numbers.

If the experiment were repeated 400 times instead of 40 times, the experimental probability would likely be even closer to \(\frac{3}{4}\).

How to Tell Whether Results Support the Law of Large Numbers

When you look at results from a small number of trials and a large number of trials, ask:

1. What is the theoretical probability?
2. What is the experimental probability for the smaller sample?
3. What is the experimental probability for the larger sample?
4. Which one is closer to the theoretical probability?

If the larger sample is closer, that supports the Law of Large Numbers.

Example:

The theoretical probability of rolling an even number on a fair 6-sided number cube is:

Even numbers are 2, 4, and 6, so there are 3 favorable outcomes out of 6.

$$P(\text{even})=\frac{3}{6}=\frac{1}{2}$$

Now compare two experiments:

• In 10 rolls, an even number appears 7 times: $$\frac{7}{10}=0.7$$
• In 100 rolls, an even number appears 52 times: $$\frac{52}{100}=0.52$$

The theoretical probability is \(0.5\). The result from 100 rolls, \(0.52\), is closer to \(0.5\) than the result from 10 rolls, \(0.7\). This supports the Law of Large Numbers.

Important Things to Remember

• Theoretical probability is based on what should happen.
• Experimental probability is based on what does happen.
• They may not match in a small experiment.
• As the number of trials increases, the experimental probability usually gets closer to the theoretical probability.
• This idea is called the Law of Large Numbers.

Common Mistakes

• Mixing up the formulas. Remember: theoretical uses possible outcomes, experimental uses actual results.
• Thinking small experiments must match theory. They do not have to match exactly.
• Thinking “closer” means “equal.” The experimental probability may get close, but it may not be exactly the same.
• Forgetting total trials. In experimental probability, the denominator is the total number of times the experiment was done.

Quick Check

Try these on your own:

1. A fair coin is flipped 20 times and lands on tails 9 times. What are the theoretical and experimental probabilities of tails?
2. A fair 6-sided number cube is rolled 60 times and lands on 2 a total of 11 times. Compare the theoretical and experimental probabilities.
3. A spinner has 4 equal sections: red, blue, green, and yellow. It is spun 80 times and lands on green 18 times. Is the experimental probability close to the theoretical probability?

Answers to Quick Check

1. Theoretical: \(\frac{1}{2}\). Experimental: \(\frac{9}{20}=0.45\).
2. Theoretical: \(\frac{1}{6}\approx 0.167\). Experimental: \(\frac{11}{60}\approx 0.183\). They are close.
3. Theoretical: \(\frac{1}{4}=0.25\). Experimental: \(\frac{18}{80}=0.225\). Yes, it is fairly close.

Summary

Probability can be found in two ways. Theoretical probability is based on the possible outcomes, and experimental probability is based on actual results from trials.

When you compare them, they may be different at first, especially with only a few trials. But as more trials are done, the experimental probability usually gets closer to the theoretical probability. This is the Law of Large Numbers.

Mutually Exclusive Events and the Addition Rule

Probability helps us describe how likely something is to happen. Sometimes we want to know the chance of one event OR another event happening.

For example, if you roll a number cube, what is the probability of rolling a 2 or a 5? Questions like this use the Addition Rule.

In this lesson, you will learn what mutually exclusive events are and how to use the Addition Rule to find probabilities.

1. What is a mutually exclusive event?

Two events are mutually exclusive if they cannot happen at the same time.

That means if one event happens, the other event does not happen in that same trial.

• When rolling one number cube, getting a 3 and getting a 6 are mutually exclusive.
• When picking one card, getting a heart and getting a spade are mutually exclusive.
• When flipping one coin, getting heads and getting tails are mutually exclusive.

These events do not overlap. There is no outcome that belongs to both events.

2. What does OR mean in probability?

In probability, the word or means that either event can happen.

If the events are mutually exclusive, we can find the probability of one event or the other by adding their probabilities.

This is called the Addition Rule for mutually exclusive events:

Here:

• \(P(A)\) means the probability of event A

• \(P(B)\) means the probability of event B

• \(P(A \text{ or } B)\) means the probability that A happens, or B happens

3. Why can we add the probabilities?

We can add the probabilities because the events do not happen together. There is no overlap to worry about.

Think about rolling one number cube:

• Event A: rolling a 1
• Event B: rolling a 4

The possible outcomes are \(1,2,3,4,5,6\).

If we want the probability of rolling a 1 or a 4, the favorable outcomes are just \(1\) and \(4\).

There are 2 favorable outcomes out of 6 total outcomes, so:

We also get the same answer by adding:

4. Steps for using the Addition Rule

1. Decide whether the events are mutually exclusive.

2. Find the probability of each event.

3. Add the probabilities.

4. Simplify your answer if needed.

5. Worked Examples

Example 1: Rolling a number cube

What is the probability of rolling a 2 or a 6 on a fair six-sided number cube?

Step 1: Check if the events are mutually exclusive.

On one roll, you cannot get both 2 and 6. So yes, they are mutually exclusive.

Step 2: Find each probability.

Step 3: Add.

Step 4: Simplify.

Answer: The probability is \(\frac{1}{3}\).

Example 2: Choosing a card by suit

A standard deck has 52 cards. What is the probability of choosing a heart or a club?

Step 1: Check if the events are mutually exclusive.

A card cannot be both a heart and a club at the same time. So the events are mutually exclusive.

Step 2: Find each probability.

There are 13 hearts and 13 clubs in a 52-card deck.

Step 3: Add.

Step 4: Simplify.

Answer: The probability is \(\frac{1}{2}\).

Example 3: Picking from a bag

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is picked at random. What is the probability of picking a blue marble or a green marble?

Step 1: Find the total number of marbles.

Step 2: Check if the events are mutually exclusive.

One marble cannot be both blue and green. So yes, the events are mutually exclusive.

Step 3: Find each probability.

Step 4: Add.

Step 5: Simplify.

Answer: The probability is \(\frac{1}{2}\).

Example 4: Using words to think carefully

A spinner has 8 equal sections labeled 1 through 8. What is the probability of landing on an odd number or landing on 6?

Step 1: List the outcomes for each event.

• Odd numbers: \(1,3,5,7\)
• Landing on 6: \(6\)

Step 2: Check if the events are mutually exclusive.

The number 6 is not odd, so there is no overlap. The events are mutually exclusive.

Step 3: Find each probability.

Step 4: Add.

Answer: The probability is \(\frac{5}{8}\).

6. A quick way to check your thinking

Ask yourself this question:

Can both events happen in the same single outcome?

• If no, the events are mutually exclusive, so you can add the probabilities.
• If yes, then they are not mutually exclusive, and you should not just add without thinking more carefully.

In this lesson, we are focusing on the first case: events that cannot happen together.

7. Common mistakes to avoid

• Adding events that are not mutually exclusive. Always check whether the events can happen at the same time.

• Using the wrong total number of outcomes. Make sure your denominator is the total number of possible outcomes.

• Forgetting to simplify. For example, \(\frac{2}{6}\) should be simplified to \(\frac{1}{3}\).

• Misreading OR. In probability, OR means one event, the other event, or whichever one happens.

8. Practice ideas

Try these on your own:

• On a six-sided number cube, what is the probability of rolling a 1 or a 3?

• From a deck of cards, what is the probability of drawing a diamond or a spade?

• A bag has 4 yellow, 6 black, and 5 white beads. What is the probability of choosing a yellow or white bead?

9. Summary

Mutually exclusive events are events that cannot happen at the same time.

When two events are mutually exclusive, you can use the Addition Rule:

This rule helps you find the probability of OR situations quickly and correctly. Just make sure the events do not overlap before you add.

Independent vs. Dependent Events

In probability, an event is something that can happen, like rolling a 4 on a number cube or picking a red marble from a bag.

Sometimes we look at more than one event. When that happens, we need to ask an important question: Does the first event change the probability of the second event?

This is how we tell the difference between independent events and dependent events.

Independent events are events where one event does not affect the other event.

Dependent events are events where one event does affect the other event.

One of the biggest clues is whether there is replacement or no replacement.

• With replacement: the item is put back, so the total number of items stays the same. The events are usually independent.
• Without replacement: the item is not put back, so the total number of items changes. The events are usually dependent.

Why this matters

When two events happen one after another, the probability of the second event may stay the same or may change.

If the probability stays the same, the events are independent. If the probability changes, the events are dependent.

To find the probability of two events happening, we multiply:

But we must be careful. For dependent events, the probability of the second event may be different after the first event happens.

How to recognize independent events

• One event does not change the outcome space of the other.
• The total number of possible outcomes stays the same.
• Examples often involve flipping coins, spinning spinners, or drawing with replacement.

How to recognize dependent events

• The first event changes what is available for the second event.
• The total number of possible outcomes changes.
• Examples often involve drawing items from a bag without replacement.

Worked Example 1: Independent events with a coin and a number cube

A coin is flipped and a number cube is rolled. What is the probability of getting heads and then a 3?

The probability of heads is:

The probability of rolling a 3 is:

These events are independent because flipping the coin does not change the roll of the number cube.

Multiply the probabilities:

So, the probability is \(\frac{1}{12}\).

Worked Example 2: Independent events with replacement

A bag has 5 red marbles and 3 blue marbles. One marble is picked, its color is recorded, and then it is put back. Then a second marble is picked. What is the probability of picking a red marble both times?

First, find the probability of red on one pick:

Because the marble is put back, the bag still has 5 red marbles out of 8 total marbles for the second pick.

So the second probability is also:

The events are independent because the first pick does not change the bag.

Multiply:

So, the probability is \(\frac{25}{64}\).

Worked Example 3: Dependent events without replacement

A bag has 5 red marbles and 3 blue marbles. One marble is picked and not put back. Then a second marble is picked. What is the probability of picking a red marble both times?

The probability of red on the first pick is:

If the first marble was red and it is not replaced, then there are now 4 red marbles left out of 7 total marbles.

So the probability of red on the second pick is now:

The events are dependent because the first pick changes the second pick.

Multiply:

So, the probability is \(\frac{5}{14}\).

Notice that this answer is different from the example with replacement. That is because without replacement changes the sample space.

Worked Example 4: Deciding whether events are independent or dependent

For each situation, decide whether the events are independent or dependent.

1. Rolling a number cube twice
2. Choosing two cards from a deck without replacement
3. Spinning a spinner and then flipping a coin
4. Choosing a marble, putting it back, and choosing again

Answers:

• Rolling a number cube twice: Independent. The first roll does not change the second roll.
• Choosing two cards without replacement: Dependent. The first card changes what cards are left.
• Spinning a spinner and flipping a coin: Independent. One does not affect the other.
• Choosing a marble, putting it back, and choosing again: Independent. The bag stays the same.

A quick way to think about it

• Ask: Did the first event change what can happen next?
• If no, the events are probably independent.
• If yes, the events are probably dependent.

Comparing independent and dependent events

• Independent: probability stays the same from one event to the next.
• Dependent: probability changes because the outcome space changes.
• Replacement: usually means independent.
• No replacement: usually means dependent.

Common mistakes to avoid

• Do not assume all two-step events are independent.
• Always check whether the first event changes the second one.
• When there is no replacement, make sure both the number of wanted outcomes and the total number of outcomes are updated.

Let’s review with one more comparison

A bag has 2 green and 6 yellow marbles.

If you pick a green marble, replace it, and pick again:

If you pick a green marble, do not replace it, and pick again:

The second probability is smaller because after the first green marble is removed, there is only 1 green marble left.

Summary

Independent events do not affect each other. Dependent events do affect each other.

When events happen with replacement, they are usually independent because the total stays the same. When events happen without replacement, they are usually dependent because the total changes.

To solve probability problems with two events, multiply the probabilities. Just remember to check whether the second probability stays the same or changes.

Multiplication Rule for Compound Events

In probability, a compound event is an event made of two or more simple events.

When a question uses the word “and”, we often need the multiplication rule. This rule helps us find the probability that multiple events happen together.

For example:

• rolling a 2 and then a 5
• picking a red marble and then a blue marble
• flipping heads and rolling an even number

The main idea is:

$$\text{Probability of event A and event B} = \text{Probability of A} \times \text{Probability of B}$$

But there is an important question to ask first:

Does the first event change the second event?

If the answer is no, the events are called independent.

If the answer is yes, the events are called dependent.

We use multiplication in both cases, but for dependent events we must be careful because the second probability may change.

1. Independent Events

Independent events are events where one event does not affect the other.

Examples of independent events include:

• flipping a coin and rolling a die
• rolling one die, then rolling it again
• spinning a spinner twice when the spinner stays the same each time

For independent events, the multiplication rule is:

$$P(A \text{ and } B) = P(A) \times P(B)$$

Example 1: Flip a coin and roll a die

What is the probability of flipping heads and rolling a 4?

Step 1: Find each probability.

• Probability of heads: \(\frac{1}{2}\)
• Probability of rolling a 4: \(\frac{1}{6}\)

Step 2: Multiply.

$$\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

So, the probability of flipping heads and rolling a 4 is \(\frac{1}{12}\).

Example 2: Roll a die twice

What is the probability of rolling an even number on the first roll and an odd number on the second roll?

Step 1: Find each probability.

• Even numbers on a die: 2, 4, 6, so \(\frac{3}{6} = \frac{1}{2}\)
• Odd numbers on a die: 1, 3, 5, so \(\frac{3}{6} = \frac{1}{2}\)

Step 2: Multiply.

$$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

So, the probability is \(\frac{1}{4}\).

2. Dependent Events

Dependent events are events where the first event changes the second event.

This often happens when choosing objects without replacement. That means you do not put the first object back before choosing again.

Because one object is removed, the total number of objects changes. That means the second probability may be different from the first.

For dependent events, we still multiply, but we use the updated probability for the second event:

$$P(A \text{ and } B) = P(A) \times P(B \text{ after } A)$$

Example 3: Choosing marbles without replacement

A bag has 3 red marbles and 2 blue marbles. What is the probability of choosing a red marble and then a blue marble without replacement?

Step 1: Find the probability of choosing red first.

There are 5 marbles total, and 3 are red.

$$P(\text{red first}) = \frac{3}{5}$$

Step 2: Find the probability of choosing blue second.

After taking one red marble, 4 marbles are left. The number of blue marbles is still 2.

$$P(\text{blue second}) = \frac{2}{4} = \frac{1}{2}$$

Step 3: Multiply.

$$\frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$$

So, the probability is \(\frac{3}{10}\).

Example 4: Two cards without replacement

You have 4 cards: 2 stars and 2 hearts. You pick 2 cards without replacement. What is the probability of picking a star and then another star?

Step 1: First pick is a star.

$$P(\text{first star}) = \frac{2}{4} = \frac{1}{2}$$

Step 2: Second pick is a star.

After one star is removed, 3 cards are left and only 1 star remains.

$$P(\text{second star}) = \frac{1}{3}$$

Step 3: Multiply.

$$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$$

So, the probability of picking 2 stars in a row is \(\frac{1}{6}\).

How to Know Which Rule Situation You Have

Use these questions:

• Are the events connected by “and”? If yes, multiplication may be needed.
• Does the first event change the second event?
• Is the item put back, or is the die/coin reset? If yes, the events are usually independent.
• Is something removed and not replaced? If yes, the events are usually dependent.

Important Notes

• Multiply probabilities for AND events.
• For independent events, the probabilities stay the same.
• For dependent events, the second probability may change.
• Always simplify your fraction if possible.

Common Mistakes to Avoid

• Adding instead of multiplying: For “and,” use multiplication, not addition.
• Forgetting to change the total: In dependent events, the total number of choices may get smaller.
• Forgetting to change the number of successful outcomes: If an item is removed, the number you want may also change.

Quick Check

1. What is the probability of rolling a 3 and then rolling a 3 again?
   Answer: $$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$
2. A bag has 5 green counters and 1 yellow counter. What is the probability of picking green and then yellow without replacement?
   Answer: $$\frac{5}{6} \times \frac{1}{5} = \frac{1}{6}$$

Summary

The multiplication rule helps you find the probability of compound events when both events must happen.

If the events are independent, multiply the two probabilities as they are.

If the events are dependent, multiply the first probability by the changed second probability.

Whenever you see “and”, think: Should I multiply?

Simulations and Random Number Models help us study chance events by acting them out many times.

Sometimes a real situation is hard to test directly. For example, it may take too long, cost too much, or be impossible to repeat exactly. A simulation is a model that copies the important parts of a real event using tools like dice, spinners, coins, cards, or random numbers.

In this lesson, you will learn how to build a simulation, how to connect random numbers to outcomes, and how to use results from a simulation to estimate probability.

Key idea: A good simulation matches the chances in the real situation as closely as possible.

What is a simulation?

A simulation is a repeated experiment that represents a real-world event. Each trial in the simulation stands for one real event.

• Rolling a die can represent choosing one of 6 equally likely outcomes.
• Flipping a coin can represent two equally likely outcomes.
• A random number generator can represent many different outcomes.

After many trials, the results from the simulation can help us estimate what is likely to happen in real life.

Why use random number models?

A random number model assigns numbers to outcomes. Then we use random digits or random numbers to choose outcomes fairly.

This is helpful when:

• the event has more than 2 outcomes,
• the outcomes are not all equally likely,
• we want to repeat the event many times quickly.

For example, if 70% of students ride the bus and 30% walk home, a simulation should give “bus” more often than “walk.”

Steps for building a simulation

1. Understand the real situation. List the possible outcomes and their chances.
2. Choose a random tool. Use a die, spinner, coin, or random numbers.
3. Assign outcomes to the tool. Make sure the model matches the real probabilities.
4. Run many trials. Each trial should copy one event.
5. Record results. Count how often each outcome happens.
6. Compare and estimate. Use the results to make predictions.

Matching the simulation to the probability

The most important part of a simulation is making sure the model is fair and matches the real chances.

If outcomes are equally likely, the model is simple. If outcomes are not equally likely, we give more random numbers to the more likely outcome.

Suppose an event has:

• Outcome A: 60%
• Outcome B: 40%

We can use random digits 0 through 9. There are 10 digits total, so 60% means 6 of the 10 digits and 40% means 4 of the 10 digits.

One possible assignment is:

• A: 0, 1, 2, 3, 4, 5
• B: 6, 7, 8, 9

Then each random digit gives one trial.

Worked Example 1: Using a die for equal outcomes

A spinner in a game has 3 equal sections: red, blue, and green. Design a simulation using a number cube labeled 1 to 6.

Step 1: The 3 colors are equally likely.

Step 2: A die has 6 equally likely results, so we can split the 6 numbers equally among the 3 colors.

Step 3: Assign outcomes.

• Red: 1, 2
• Blue: 3, 4
• Green: 5, 6

Why this works: Each color gets 2 of the 6 numbers, so each has probability

$$\frac{2}{6}=\frac{1}{3}$$

This matches the spinner.

Worked Example 2: Using random digits for unequal outcomes

A weather forecast says there is a 70% chance of sunshine and a 30% chance of rain tomorrow. Create a random number model.

Step 1: Use random digits 0 to 9. There are 10 possible digits.

Step 2: Assign 7 digits to sunshine and 3 digits to rain, because

$$70\%=\frac{7}{10} \qquad 30\%=\frac{3}{10}$$

One possible model:

• Sunshine: 0, 1, 2, 3, 4, 5, 6
• Rain: 7, 8, 9

How to use it: If a random digit is 5, record sunshine. If the digit is 8, record rain.

If we generate many random digits, the percent of sunshine results should be close to 70%.

Worked Example 3: Simulating a real-world situation with two events

A school survey shows that 40% of students bring lunch from home and 60% buy lunch. Design a simulation for 5 students, then use the random digits below:

Random digits: 2, 9, 5, 1, 7

Step 1: Use digits 0 to 9.

Step 2: Since 40% = 4 out of 10, assign 4 digits to “bring” and 6 digits to “buy.”

• Bring from home: 0, 1, 2, 3
• Buy lunch: 4, 5, 6, 7, 8, 9

Step 3: Read each digit.

• 2 → bring
• 9 → buy
• 5 → buy
• 1 → bring
• 7 → buy

Result: Out of 5 students in the simulation:

• 2 bring lunch
• 3 buy lunch

This does not match exactly 40% and 60%, but that is normal in a small sample. Simulations usually get closer to the expected probabilities when we do more trials.

Worked Example 4: Simulating a compound event

A game is played by flipping a coin and rolling a number cube. You win if you get heads and an even number. How can a simulation model this game?

Step 1: List the parts of the event.

• Coin: heads or tails
• Die: 1, 2, 3, 4, 5, 6

Step 2: One trial means doing both random actions once.

You could:

• flip a real coin and roll a real die, or
• use random numbers.

One random number model:

• Use a random digit for the coin: 0–4 = heads, 5–9 = tails
• Use a die roll for the number: 1 to 6

Winning outcomes: heads with 2, 4, or 6.

The probability of heads is

$$\frac{1}{2}$$

The probability of an even number is

$$\frac{3}{6}=\frac{1}{2}$$

So the probability of winning is

$$\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$$

That means in many simulation trials, we expect about 1 out of every 4 games to be wins.

Comparing theoretical probability and experimental probability

Theoretical probability is what math predicts should happen.

Experimental probability is what actually happens in a test or simulation.

We can write experimental probability as

$$\text{experimental probability}=\frac{\text{number of times the event happened}}{\text{total number of trials}}$$

For example, if a simulation of 20 days gives rain on 7 days, then the experimental probability of rain is

$$\frac{7}{20}$$

As the number of trials increases, the experimental probability often gets closer to the theoretical probability.

How to tell if a simulation is good

A good simulation should:

• match the real outcomes,
• match the real probabilities,
• use a fair random process,
• have enough trials to make the results useful.

Example of a poor simulation: If an event has a 75% chance of success, using a coin would not be a good model because a coin only gives 50% and 50%.

Better model: Use random digits 0 to 9, with 0–7 as success and 8–9 as failure. That gives 8 out of 10, or 80%, which is closer. Even better, if possible, use 100 random numbers and assign 75 of them to success.

Common mistakes to avoid

• Not matching the probabilities correctly. Check that the number assignments fit the percentages or fractions.
• Using too few trials. A small sample can be misleading.
• Forgetting what one trial means. One trial must represent one full event.
• Using an unfair tool. A biased spinner or loaded die gives bad results.

Helpful strategy

When you design a simulation, ask yourself these questions:

1. What are the outcomes?
2. Are they equally likely?
3. What random tool should I use?
4. How do I assign numbers or sections fairly?
5. What does one trial represent?

Brief Summary

A simulation is a way to model a chance event using random tools such as dice, coins, spinners, or random numbers. A random number model works by assigning numbers to outcomes so the model matches the real probabilities. After many trials, the simulation results can be used to estimate how likely events are, especially in real-world situations and compound events.