Probability and Compound Events

Theoretical vs. Experimental Probability

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

There are two important ways to think about probability:

• Theoretical probability — what we expect to happen based on math.
• Experimental probability — what actually happens when we do an experiment or collect data.

Understanding the difference between these two ideas helps us compare expected results with real-life results.

1. Theoretical Probability

Theoretical probability is found before doing an experiment. It is based on all the possible outcomes, assuming each outcome is equally likely.

The formula is:

A favorable outcome is an outcome you want to happen.

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

If you want the probability of rolling a 3, there is 1 favorable outcome out of 6 possible outcomes.

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

2. Experimental Probability

Experimental probability is found after doing an experiment. It uses the results from actual trials.

The formula is:

A trial is one time you perform the experiment.

For example, suppose you flip a coin 20 times and it lands on heads 13 times.

Then the experimental probability of getting heads is:

This does not match the theoretical probability exactly, even though the theoretical probability of heads on a fair coin is:

This difference can happen because real experiments do not always match the expected results perfectly, especially when the number of trials is small.

3. Key Difference Between Theoretical and Experimental Probability

The main difference is:

• Theoretical probability is based on what should happen mathematically.
• Experimental probability is based on what did happen in actual trials.

Here is a quick comparison:

• Theoretical probability uses possible outcomes.
• Experimental probability uses observed results.
• Theoretical probability is calculated without testing.
• Experimental probability is calculated after testing.

4. Why Are They Sometimes Different?

Even if an experiment is fair, the results may not match the theoretical probability exactly.

For example, a fair coin has a theoretical probability of \(\frac{1}{2}\) for heads. But if you flip it 4 times, you might get 3 heads and 1 tails.

The experimental probability of heads would then be:

That is different from \(\frac{1}{2}\). This does not mean the coin is unfair. It just means that with only a few trials, results can vary.

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

For example:

• In 10 coin flips, heads might appear 7 times.
• In 100 coin flips, heads might appear 52 times.
• In 1,000 coin flips, heads might appear 498 times.

The more trials you do, the more stable the results tend to become.

5. How to Find Theoretical Probability

1. List all possible outcomes.
2. Count how many outcomes are favorable.
3. Write the probability as a fraction.
4. Simplify if possible.

Example: A bag has 5 red marbles, 3 blue marbles, and 2 green marbles. What is the theoretical probability of choosing a blue marble?

Total marbles:

Favorable outcomes: 3 blue marbles

6. How to Find Experimental Probability

1. Do the experiment or use given data.
2. Count how many times the event occurred.
3. Count the total number of trials.
4. Write the probability as a fraction, decimal, or percent.

Example: A spinner is spun 40 times. It lands on blue 14 times. What is the experimental probability of landing on blue?

As a decimal, this is:

As a percent, this is:

7. Worked Examples

Example 1: Simple Theoretical Probability

A fair number cube is rolled once. What is the theoretical probability of rolling a number greater than 4?

Step 1: List the possible outcomes.

Possible outcomes: 1, 2, 3, 4, 5, 6

Step 2: Find the favorable outcomes.

Numbers greater than 4 are 5 and 6, so there are 2 favorable outcomes.

Step 3: Write the probability.

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

Example 2: Simple Experimental Probability

A coin is flipped 30 times, and tails comes up 17 times. What is the experimental probability of getting tails?

Use the formula:

Substitute the values:

Answer: The experimental probability is \(\frac{17}{30}\).

Example 3: Compare Theoretical and Experimental Probability

A bag contains 4 yellow marbles and 6 purple marbles. A marble is chosen, replaced, and the experiment is repeated 50 times. Yellow is selected 23 times.

Find:

• the theoretical probability of choosing yellow
• the experimental probability of choosing yellow

Step 1: Find the theoretical probability.

Total marbles:

Favorable outcomes: 4 yellow marbles

Step 2: Find the experimental probability.

Yellow was chosen 23 times in 50 trials.

Step 3: Compare them.

The theoretical probability is \(\frac{2}{5} = 0.4\).

The experimental probability is \(\frac{23}{50} = 0.46\).

These values are close, but not exactly the same. That is normal in experiments.

Answer:

• Theoretical probability: \(\frac{2}{5}\)
• Experimental probability: \(\frac{23}{50}\)

Example 4: Using Data to Predict

A spinner is spun 60 times. It lands on red 18 times. Based on the experimental probability, predict how many times it will land on red in 100 spins.

Step 1: Find the experimental probability.

Step 2: Use that probability to predict the number of reds in 100 spins.

Answer: Based on the experimental probability, we would predict about 30 red spins out of 100.

8. Important Ideas to Remember

• Probability 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.
• All probabilities are between 0 and 1.
• Theoretical probability is based on math and possible outcomes.
• Experimental probability is based on actual results.
• More trials usually make experimental probability closer to theoretical probability.

9. Common Mistakes

• Mixing up the formulas: Theoretical uses possible outcomes, while experimental uses actual results.
• Forgetting the total number of outcomes: Count all possible outcomes carefully.
• Using too few trials to make a big conclusion: Small samples can give unusual results.
• Not simplifying fractions: For example, \(\frac{3}{6}\) should be simplified to \(\frac{1}{2}\).

10. Quick Check for Understanding

Try these on your own:

1. A fair coin is flipped once. What is the theoretical probability of heads?
2. A number cube is rolled 24 times, and a 6 appears 5 times. What is the experimental probability of rolling a 6?
3. A bag has 7 black marbles and 3 white marbles. What is the theoretical probability of choosing a white marble?
4. A spinner lands on green 9 times out of 25 spins. What is the experimental probability of green?

Answers:

1. \(\frac{1}{2}\)
2. \(\frac{5}{24}\)
3. \(\frac{3}{10}\)
4. \(\frac{9}{25}\)

Summary

Theoretical probability tells what is expected to happen based on all possible outcomes. Experimental probability tells what actually happened during trials or experiments.

They may not always be equal, especially when the number of trials is small. But as the number of trials increases, the experimental probability often gets closer to the theoretical probability.

When solving problems, always ask yourself: Am I using expected outcomes, or am I using real data? That will help you choose the correct type of probability.

The Law of Large Numbers is an important idea in probability. It helps us understand what happens when we repeat the same chance experiment many times.

At first, results can look surprising. If you flip a coin only a few times, you might get more heads than tails. But if you keep flipping the coin many, many times, the results usually get closer to what probability predicts.

This idea is called the Law of Large Numbers: as the number of trials increases, the experimental probability gets closer to the theoretical probability.

To understand this well, we need to know the difference between theoretical probability and experimental probability.

Theoretical probability is what we expect to happen based on math, before doing the experiment.

It is calculated by

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

Experimental probability is what actually happens when we do the experiment.

It is calculated by

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

The Law of Large Numbers says that these two probabilities become closer when the number of trials becomes very large.

Why does this happen?

Chance experiments can be uneven in the short term. In a small number of trials, luck can have a big effect.

But over many trials, the ups and downs begin to balance out. The results become more stable, and the experimental probability usually moves closer to the theoretical probability.

This does not mean the results will be exactly equal every time. It means they usually get closer.

Example: Flipping a fair coin

A fair coin has two equally likely outcomes: heads and tails.

So the theoretical probability of getting heads is

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

Now imagine these results:

• After 10 flips, heads happened 7 times.
• After 100 flips, heads happened 54 times.
• After 1,000 flips, heads happened 503 times.

Let us calculate the experimental probability each time.

After 10 flips:

$$\frac{7}{10} = 0.7$$

After 100 flips:

$$\frac{54}{100} = 0.54$$

After 1,000 flips:

$$\frac{503}{1000} = 0.503$$

The theoretical probability is

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

Notice what happens:

• 0.7 is not very close to 0.5
• 0.54 is closer
• 0.503 is even closer

This is the Law of Large Numbers in action.

Main idea to remember

Small samples can be misleading. Large samples give better estimates.

If you only do an experiment a few times, your results may not match the theoretical probability very well. But if you repeat it many times, the results usually become more reliable.

Worked Example 1: Rolling a number cube

A fair number cube has 6 equally likely outcomes: 1, 2, 3, 4, 5, and 6.

What is the theoretical probability of rolling a 4?

There is 1 favorable outcome out of 6 possible outcomes.

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

Suppose a student rolls the number cube 12 times and gets a 4 three times.

The experimental probability is

$$\frac{3}{12} = \frac{1}{4} = 0.25$$

The theoretical probability is

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

These are not equal, and that is okay. With only 12 trials, the experimental probability can be quite different.

If the student rolls the cube 600 times and gets a 4 exactly 98 times, then the experimental probability is

$$\frac{98}{600} \approx 0.163$$

Now it is much closer to

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

This shows that more trials usually give results closer to the theoretical probability.

Worked Example 2: Choosing a colored marble

A bag contains 3 red marbles and 2 blue marbles. One marble is chosen at random, then replaced, and the experiment is repeated.

What is the theoretical probability of choosing a red marble?

There are 3 red marbles out of 5 total marbles.

$$P(\text{red}) = \frac{3}{5} = 0.6$$

Suppose in 10 trials, red is chosen 8 times.

The experimental probability is

$$\frac{8}{10} = 0.8$$

This is higher than 0.6.

Now suppose in 200 trials, red is chosen 121 times.

The experimental probability is

$$\frac{121}{200} = 0.605$$

This is very close to 0.6.

Again, the larger number of trials gives an experimental probability closer to the theoretical probability.

Worked Example 3: Spinner experiment

A spinner is divided into 4 equal sections: 1 green, 1 yellow, 1 blue, and 1 red.

What is the theoretical probability of landing on blue?

Since the sections are equal,

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

Suppose the spinner is spun 20 times and lands on blue 2 times.

The experimental probability is

$$\frac{2}{20} = 0.1$$

That is quite different from 0.25.

Now suppose the spinner is spun 400 times and lands on blue 96 times.

The experimental probability is

$$\frac{96}{400} = 0.24$$

This is much closer to 0.25.

So even though small numbers of spins can give unusual results, many spins usually give results closer to the expected probability.

Important facts about the Law of Large Numbers

• It does not guarantee exact results. Even after many trials, experimental probability may not be exactly the same as theoretical probability.
• It describes a trend. As the number of trials increases, results tend to get closer to the expected value.
• It works for many kinds of probability experiments. Coins, dice, spinners, cards, and marbles can all show this pattern.
• It helps explain fairness. If an experiment is fair, large numbers of trials can show whether the results match what is expected.

A common misunderstanding

Some students think that if a coin lands on heads many times in a row, then tails is "due" next. That is not what the Law of Large Numbers means.

Each flip of a fair coin still has probability

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

for heads and

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

for tails.

The Law of Large Numbers is about the overall results after many trials, not about forcing the next result to be different.

For example, if a fair coin lands on heads 6 times in a row, the next flip is still just as likely to be heads as tails.

How this connects to compound events

In probability, a compound event is an event made of two or more simple events. For example, flipping two coins and getting two heads is a compound event.

The Law of Large Numbers also applies to compound events. If you repeat a compound event many times, the experimental probability should get closer to the theoretical probability.

For example, if you flip two fair coins, the possible outcomes are:

$$HH, HT, TH, TT$$

There is 1 outcome with two heads out of 4 possible outcomes.

So the theoretical probability of getting two heads is

$$P(HH) = \frac{1}{4}$$

If you try this only 8 times, you may not get exactly 2 successes. But if you try it hundreds of times, the fraction of times you get two heads will likely be closer to

$$\frac{1}{4}$$

Worked Example 4: Two-coin compound event

A student flips two fair coins 16 times. They get two heads 6 times.

The experimental probability is

$$\frac{6}{16} = \frac{3}{8} = 0.375$$

The theoretical probability of two heads is

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

These are not very close.

Now imagine the student repeats the experiment 800 times and gets two heads 198 times.

The experimental probability is

$$\frac{198}{800} = 0.2475$$

This is very close to

$$0.25$$

This larger sample better matches the theoretical probability.

How to recognize the Law of Large Numbers in a problem

You may be using the Law of Large Numbers when a problem talks about:

• repeating an experiment many times,
• comparing expected results and actual results,
• small samples versus large samples,
• experimental probability getting closer to theoretical probability.

Steps for solving problems

1. Find the theoretical probability.
   Use the number of favorable outcomes and total possible outcomes.
2. Find the experimental probability.
   Use the number of times the event happened and the total number of trials.
3. Compare the two probabilities.
   See how close they are.
4. Think about the number of trials.
   A larger number of trials usually gives a better estimate of the theoretical probability.

Quick check

Suppose the theoretical probability of an event is

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

Which experimental result better matches the theoretical probability?

• 3 successes in 5 trials: $$\frac{3}{5} = 0.6$$
• 61 successes in 200 trials: $$\frac{61}{200} = 0.305$$

The second result is much closer to 0.3, so it better matches the theoretical probability.

This is what we expect from the Law of Large Numbers.

Summary

The Law of Large Numbers says that when a chance experiment is repeated many times, the experimental probability usually gets closer to the theoretical probability.

Small numbers of trials can give results that look strange or uneven. Large numbers of trials usually give more reliable results.

Remember: the results do not have to become exactly equal. They just tend to become closer as the number of trials increases.

Sample Spaces and the Fundamental Counting Principle

When we study probability, we often want to know all the possible outcomes of an event. For example, if you flip a coin, the outcomes are heads or tails. If you roll a number cube, the outcomes are 1 through 6.

A sample space is the complete list of all possible outcomes. Knowing the sample space helps us find probabilities and understand compound events, which are events made of two or more simpler events.

In this lesson, you will learn how to:

• find a sample space,
• organize outcomes using lists, tables, and tree diagrams,
• use the Fundamental Counting Principle, and
• connect counting outcomes to probability.

1. What is a sample space?

A sample space is the set of every possible outcome of an experiment.

Examples:

• Flip 1 coin: \,\(\{H, T\}\)
• Roll 1 number cube: \,\(\{1,2,3,4,5,6\}\)
• Choose a day of the weekend: \,\(\{Saturday, Sunday\}\)

If an event has more than one step, the sample space includes all combinations of those steps.

For example, if you flip a coin and then roll a number cube, the outcomes are:

\(\{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6\}\)

There are 12 outcomes in that sample space.

2. Ways to generate a sample space

There are several useful ways to organize all possible outcomes:

• List
• Table
• Tree diagram
• Multiplication using the Fundamental Counting Principle

Each method helps you make sure you do not miss any outcomes or count any outcome twice.

3. Making an organized list

An organized list works well when there are not too many outcomes. The key is to use a pattern.

Worked Example 1: Flip a coin twice

Find the sample space for flipping a coin two times.

Possible outcomes from the first flip: \,\(H\) or \,\(T\)

Possible outcomes from the second flip: \,\(H\) or \,\(T\)

List the outcomes in order:

\(\{HH, HT, TH, TT\}\)

So the sample space has:

$$4 \text{ outcomes}$$

Notice that \,\(HT\) and \,\(TH\) are different because the order matters. The first letter shows the first flip, and the second letter shows the second flip.

4. Using a table

A table is helpful when combining two sets of outcomes. One event can go across the top, and the other can go down the side.

Worked Example 2: Roll a number cube and spin a spinner with colors red, blue, and green

The number cube has outcomes \,\(1,2,3,4,5,6\).

The spinner has outcomes Red, Blue, Green.

We can think of the sample space as pairs:

\((1, R), (1, B), (1, G), (2, R), (2, B), (2, G)\), and so on.

To count them, notice:

• 6 possible number outcomes
• 3 possible color outcomes

So there are:

$$6 \times 3 = 18$$

possible outcomes.

If we listed them all, the sample space would have 18 ordered pairs.

5. Using a tree diagram

A tree diagram shows choices step by step. Each branch stands for one possible outcome at that step.

This is especially useful for compound events with two or three stages.

Worked Example 3: Choose an outfit

A student has:

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

How many different outfits are possible?

Using a tree diagram idea:

• From red shirt, you can choose black, gray, or tan pants.
• From blue shirt, you can choose black, gray, or tan pants.

List the outfits:

Red-black, Red-gray, Red-tan, Blue-black, Blue-gray, Blue-tan

There are:

$$6 \text{ outfits}$$

This example also leads us to an important shortcut.

6. The Fundamental Counting Principle

The Fundamental Counting Principle says:

If one event can happen in \,\(a\) ways and another event can happen in \,\(b\) ways, then the two events together can happen in

$$a \times b$$

ways.

If there are more than two steps, multiply all the numbers of choices together.

For example, if there are 2 choices for one step, 3 choices for another step, and 4 choices for a third step, then the total number of outcomes is:

$$2 \times 3 \times 4 = 24$$

This principle helps you count quickly without listing every outcome, especially when the sample space is large.

7. When should you list, draw, or multiply?

• Use an organized list when there are only a few outcomes.
• Use a table when combining two groups of outcomes.
• Use a tree diagram when you want to see each step clearly.
• Use the Fundamental Counting Principle when there are many outcomes and you only need the total number.

Often, these methods give the same answer. The best method depends on the problem.

8. A more challenging counting example

Worked Example 4: Create a meal

A restaurant offers:

• 3 sandwiches
• 2 side dishes
• 4 drinks

If you choose one sandwich, one side dish, and one drink, how many different meals can you make?

There are 3 choices for the sandwich.

For each sandwich, there are 2 choices for the side dish.

For each sandwich-and-side combination, there are 4 drink choices.

Use the Fundamental Counting Principle:

$$3 \times 2 \times 4 = 24$$

So, there are:

$$24 \text{ different meals}$$

9. Connecting sample spaces to probability

Once you know the sample space, you can find probability.

The theoretical probability of an event is:

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

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

\(\{HH, HT, TH, TT\}\)

Suppose you want the probability of getting exactly one head.

The favorable outcomes are \,\(HT\) and \,\(TH\).

That is 2 favorable outcomes out of 4 total outcomes.

So the probability is:

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

This shows why sample spaces matter: they help you count all outcomes fairly and correctly.

10. Common mistakes to avoid

• Forgetting order matters: In some experiments, \,\(HT\) is different from \,\(TH\).
• Missing outcomes: If your list is not organized, you may leave some out.
• Counting outcomes twice: Be careful not to repeat combinations.
• Adding instead of multiplying: If you are combining steps, you usually multiply the number of choices.

11. Quick check for understanding

Ask yourself:

• Did I include every possible outcome?
• Did I keep the outcomes organized?
• Does order matter in this situation?
• Should I list the outcomes, draw a diagram, or multiply?

Summary

A sample space is the complete set of all possible outcomes. You can generate sample spaces by making organized lists, tables, or tree diagrams. When an event has several steps, the Fundamental Counting Principle helps you find the total number of outcomes by multiplying the number of choices at each step.

These skills are important because they help you understand compound events and calculate probability correctly. If you can find all possible outcomes in an organized way, you are building a strong foundation for probability.

Probability of Independent Events

Probability helps us describe how likely something is to happen. When we talk about independent events, we mean events where one event does not change the probability of the other event.

For example, if you flip a coin and roll a number cube, the coin result does not affect the number you roll. These are independent events.

In this lesson, you will learn how to recognize independent events and how to find the probability that both events happen.

1. Review: What is probability?

Probability is a number from 0 to 1 that tells how likely an event is.

• 0 means the event is impossible.
• 1 means the event is certain.
• A probability between 0 and 1 means the event might happen.

You can find probability with this formula:

For example, the probability of rolling a 4 on a fair number cube is:

2. What does independent mean?

Two events are independent if the outcome of one event does not affect the outcome of the other event.

Here are some examples of independent events:

• Flipping a coin and rolling a number cube
• Choosing a card, putting it back, and then choosing again
• Spinning a spinner twice, if each spin is separate

Here are some examples that are not independent:

• Choosing a card and not putting it back before choosing again
• Taking marbles out of a bag without replacing them

In those cases, the first event changes what is available in the second event.

3. The rule for independent events

To find the probability that two independent events both happen, multiply their probabilities.

This is the most important rule in this lesson.

If you want the probability of three independent events, you multiply all three probabilities:

Why do we multiply?

Think of each event as a step. To get the probability that every required step happens, we multiply the chance of each step.

For example, if an event has probability \(\frac{1}{2}\) and another has probability \(\frac{1}{6}\), then the chance that both happen is smaller than either one alone:

This makes sense because getting both events is harder than getting just one.

4. Worked Example 1: Flip a coin and roll a number cube

Question: What is the probability of flipping heads and rolling a 5?

Step 1: Find the probability of heads.

Step 2: Find the probability of rolling a 5.

Step 3: Multiply because the events are independent.

Answer: The probability is \(\frac{1}{12}\).

5. Worked Example 2: Two spins on a spinner

Question: A spinner has 4 equal sections: red, blue, green, and yellow. What is the probability of landing on red twice in a row?

Each spin is independent, because one spin does not change the next spin.

Step 1: Find the probability of red on one spin.

Step 2: Multiply for two spins.

Answer: The probability is \(\frac{1}{16}\).

6. Worked Example 3: Choosing with replacement

Question: A bag has 3 blue marbles and 2 green marbles. One marble is chosen, then put back. Then a second marble is chosen. What is the probability of choosing a blue marble both times?

Because the first marble is put back, the second choice is not affected. The events are independent.

Step 1: Find the probability of blue on one draw.

There are 5 marbles total, and 3 are blue.

Step 2: Multiply for both draws.

Answer: The probability is \(\frac{9}{25}\).

7. Worked Example 4: Three independent events

Question: What is the probability of rolling an even number, flipping tails, and rolling a number greater than 4?

These are three separate events, and they are independent.

Step 1: Probability of rolling an even number on a fair number cube.

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

Step 2: Probability of flipping tails.

Step 3: Probability of rolling a number greater than 4.

The numbers greater than 4 are 5 and 6, so:

Step 4: Multiply all three probabilities.

Answer: The probability is \(\frac{1}{12}\).

8. How to tell if events are independent

Ask yourself this question:

Does the first event change the possible outcomes of the second event?

• If no, the events are probably independent.
• If yes, the events are not independent.

For example:

• Roll a die, then roll again: independent
• Pick a marble and replace it, then pick again: independent
• Pick a marble and do not replace it, then pick again: not independent

9. Common mistakes to avoid

• Adding instead of multiplying: For independent events happening together, multiply the probabilities.
• Forgetting to simplify: Reduce fractions when possible.
• Not checking independence: Make sure one event does not affect the other.
• Ignoring replacement: “Put back” or “with replacement” usually means the events are independent.

10. Quick practice thinking

Try these on your own:

1. What is the probability of flipping heads twice?
   Hint: \(\frac{1}{2} \cdot \frac{1}{2}\)
2. What is the probability of rolling a 2 and then a 6?
   Hint: \(\frac{1}{6} \cdot \frac{1}{6}\)
3. A bag has 4 red and 1 blue marble. A marble is chosen, replaced, and chosen again. What is the probability of getting blue both times?
   Hint: \(\frac{1}{5} \cdot \frac{1}{5}\)

11. Summary

Independent events are events where one outcome does not affect the other. To find the probability that independent events happen together, multiply their probabilities.

Remember the rule:

If there are more than two independent events, keep multiplying. Always check whether the events are truly independent before using this rule.

Probability of Dependent Events

Sometimes in probability, one event can change what happens next. This is called a dependent event.

Dependent events happen when the outcome of the first event affects the probability of the second event. This often happens in situations described as without replacement.

For example, if you pull a marble out of a bag and do not put it back, the total number of marbles changes. That means the probability for the next draw also changes.

In this lesson, you will learn how to recognize dependent events, how to calculate their probabilities, and how to solve problems where the sample space changes after each event.

1. Review: What is probability?

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

The basic probability formula is:

Example: If a bag has 3 red marbles and 2 blue marbles, then the probability of drawing a red marble is:

2. What makes events dependent?

Two events are dependent if the first event changes the probability of the second event.

This usually happens when:

• an item is chosen,
• it is not replaced, and
• a second item is chosen.

Because the first choice removes one item, the total number of possible outcomes becomes smaller. Sometimes the number of favorable outcomes changes too.

3. Dependent vs. independent events

It is helpful to compare dependent events with independent events.

• Independent events: The first event does not affect the second event.
• Dependent events: The first event does affect the second event.

Example of independent events: Rolling a die, then flipping a coin. The die roll does not change the coin flip.

Example of dependent events: Drawing two cards from a deck without replacement. The first card changes what is left for the second draw.

4. How to find the probability of dependent events

To find the probability of two dependent events both happening, multiply:

• the probability of the first event, and
• the probability of the second event after the first event happens.

The formula is:

This means you must adjust the second probability based on what happened first.

5. Important idea: the sample space changes

In dependent events, the sample space changes after the first event.

If there were 10 objects at first and 1 is removed, then only 9 objects remain for the second event.

This change is the key to solving dependent probability problems correctly.

6. Worked Example 1: Two red marbles

A bag contains 4 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both marbles are red?

Step 1: Find the probability of drawing a red marble first.

There are 4 red marbles out of 7 total marbles.

Step 2: Find the probability of drawing a red marble second.

After one red marble is removed, 3 red marbles remain. There are now 6 total marbles left.

Step 3: Multiply the probabilities.

Answer: The probability of drawing two red marbles is \(\frac{2}{7}\).

7. Worked Example 2: A boy then a girl

A class has 5 boys and 7 girls. Two students are chosen without replacement. What is the probability that the first student is a boy and the second student is a girl?

Step 1: First choose a boy.

There are 5 boys out of 12 students.

Step 2: Then choose a girl.

After one boy is chosen, 11 students remain. The number of girls is still 7.

Step 3: Multiply.

Answer: The probability is \(\frac{35}{132}\).

8. Worked Example 3: Two cards that are hearts

A standard deck has 52 cards. There are 13 hearts. Two cards are drawn without replacement. What is the probability that both cards are hearts?

Step 1: First heart.

Step 2: Second heart.

After one heart is removed, 12 hearts remain out of 51 cards.

Step 3: Multiply.

Simplify first if you want:

So:

Answer: The probability of drawing two hearts is \(\frac{1}{17}\).

9. Worked Example 4: Not the same color

A bag contains 6 green marbles and 4 yellow marbles. Two marbles are drawn without replacement. What is the probability that the first marble is green and the second marble is yellow?

Step 1: Probability of green first.

Step 2: Probability of yellow second.

After a green marble is removed, 9 marbles remain. The number of yellow marbles is still 4.

Step 3: Multiply.

Answer: The probability is \(\frac{4}{15}\).

10. Steps to solve dependent probability problems

1. Read carefully and look for the words without replacement.
2. Find the probability of the first event.
3. Adjust the total number of items for the second event.
4. Adjust the number of favorable outcomes if needed.
5. Multiply the probabilities.
6. Simplify your answer.

11. Common mistakes to avoid

• Forgetting to change the total: After one item is removed, the total number gets smaller.
• Forgetting to change the favorable outcomes: If the removed item is one you wanted, the number of favorable outcomes also decreases.
• Treating dependent events like independent events: Do not use the same fraction twice if the first event changes the second.

12. Quick comparison

Suppose a bag has 3 red marbles and 2 blue marbles.

If you draw two marbles with replacement, the probability of red both times is:

If you draw two marbles without replacement, the probability of red both times is:

The answers are different because the second situation is dependent.

13. Summary

Dependent events happen when one event changes the next event. This is common in problems that say without replacement.

To find the probability of dependent events, multiply the probability of the first event by the probability of the second event after the first one happens.

Always remember to update the sample space. The total number of possible outcomes gets smaller after an item is removed.

When you see a probability problem with items being chosen and not put back, think: the second probability must change.

Simulations in Probability

Sometimes in probability, it is easy to find an answer with a formula. For example, the probability of rolling a 3 on a fair number cube is \(\frac{1}{6}\).

But some situations are more complicated. Maybe there are several steps, or maybe the event is based on real-world choices. In these cases, we can use a simulation.

A simulation is a model of a real situation that uses random outcomes to estimate probability. It does not give an exact answer every time, but if you repeat it many times, it can give a good estimate.

Simulations help us answer questions like these:

• What is the chance of getting at least one head when flipping 3 coins?
• What is the chance that two students have the same birthday month?
• What is the chance of winning a game with several possible moves?

Why use a simulation?

• It helps when a problem is too complicated to solve quickly by hand.
• It shows how probability works through repeated trials.
• It lets us compare experimental probability to what we expect in theory.

Important idea: A simulation should match the real event as closely as possible. That means the random tool you use must represent the possible outcomes fairly.

For example:

• A coin can represent two equally likely outcomes.
• A number cube can represent six equally likely outcomes.
• Random digits can represent outcomes if each digit is assigned in a fair way.

Parts of a simulation

Every good simulation has a few basic parts:

1. The real event you want to study.
2. A random model to represent the event.
3. Rules for what counts as a success.
4. Many trials so the estimate becomes more reliable.

Experimental probability from a simulation is found by:

The more trials you run, the closer your experimental probability often gets to the theoretical probability.

Choosing a random model

To build a simulation, first identify all possible outcomes and how likely they are.

If outcomes are equally likely, the model is usually simple. For example, if a spinner has 4 equal sections, you can let random digits \(1,2,3,4\) represent the sections.

If outcomes are not equally likely, your model must show that too. For example, if an event has a \(70\%\) chance of happening, then on random digits from 0 to 9, you could let 7 digits mean “yes” and 3 digits mean “no.”

Example 1: Simulating a simple event

A game is won if a coin lands on heads. Use a simulation to estimate the probability of winning.

Step 1: Choose a random model.

• Heads = win
• Tails = lose

Step 2: Run trials. Suppose we flip a coin 10 times and get:

H, T, H, H, T, T, H, T, H, H

Step 3: Count successes.

• Number of wins (heads): 6
• Total trials: 10

Step 4: Find experimental probability.

The simulation estimate is \(0.6\), or \(60\%\).

This is not exactly \(\frac{1}{2}\), but with only 10 trials, that is normal. More trials would usually give a result closer to \(0.5\).

Example 2: Simulating a compound event with two number cubes

Suppose you want to estimate the probability that the sum of two number cubes is greater than 8.

Step 1: Choose a random model.

Roll two number cubes. Record the sum. A success is any sum greater than 8.

Step 2: Run some trials. Suppose these are 12 simulated rolls:

• \(3+4=7\)
• \(6+5=11\)
• \(2+2=4\)
• \(4+6=10\)
• \(1+3=4\)
• \(5+5=10\)
• \(2+6=8\)
• \(6+6=12\)
• \(3+5=8\)
• \(4+4=8\)
• \(5+2=7\)
• \(6+3=9\)

Step 3: Count successes.

Sums greater than 8 are: 11, 10, 10, 12, 9. That is 5 successes.

Step 4: Find experimental probability.

So the simulation estimate is \(\frac{5}{12}\), or about \(0.417\).

This is an estimate of the probability that the sum is greater than 8.

Example 3: Using random digits to simulate a real-world event

A weather report says there is a \(30\%\) chance of rain on a day. How can we simulate whether it rains over 10 days?

Step 1: Choose a random model.

Use random digits 0 through 9.

• Let 0, 1, 2 mean rain.
• Let 3, 4, 5, 6, 7, 8, 9 mean no rain.

This works because 3 out of 10 digits represent rain, which matches \(30\%\).

Step 2: Use random digits. Suppose the 10 digits are:

4, 1, 8, 0, 6, 2, 9, 3, 1, 7

Step 3: Decide rain or no rain.

• 4 = no rain
• 1 = rain
• 8 = no rain
• 0 = rain
• 6 = no rain
• 2 = rain
• 9 = no rain
• 3 = no rain
• 1 = rain
• 7 = no rain

Step 4: Count rainy days.

There are 4 rainy days out of 10.

Step 5: Find experimental probability.

In this simulation, rain happened on \(40\%\) of the days. That is different from \(30\%\), but with more than 10 days, the result would usually move closer to \(30\%\).

Example 4: Designing your own simulation

A school survey shows that about \(60\%\) of students prefer pizza for lunch and \(40\%\) prefer sandwiches. Design a simulation to estimate the probability that in 3 students chosen at random, at least 2 prefer pizza.

Step 1: Model one student.

Use random digits 0 through 9.

• 0, 1, 2, 3, 4, 5 = pizza
• 6, 7, 8, 9 = sandwiches

This matches the \(60\%\) and \(40\%\) chances.

Step 2: Model 3 students.

Read random digits in groups of 3. Each group is one trial.

Suppose the digits are:

5 8 2 | 1 6 4 | 7 3 0 | 9 2 8 | 4 5 1

Now check each group:

• 5 8 2 = pizza, sandwiches, pizza → 2 pizza → success
• 1 6 4 = pizza, sandwiches, pizza → 2 pizza → success
• 7 3 0 = sandwiches, pizza, pizza → 2 pizza → success
• 9 2 8 = sandwiches, pizza, sandwiches → 1 pizza → not a success
• 4 5 1 = pizza, pizza, pizza → 3 pizza → success

Step 3: Count successes.

There are 4 successes out of 5 trials.

Step 4: Find experimental probability.

The simulation estimates that the probability is \(0.8\), or \(80\%\), that at least 2 of the 3 students prefer pizza.

How to design a simulation step by step

1. Read the problem carefully.
2. Decide what outcome you want to estimate.
3. Choose a fair random tool: coin, spinner, number cube, or random digits.
4. Assign each possible outcome to part of the random tool.
5. Clearly define what counts as a success.
6. Run many trials.
7. Use the results to find experimental probability.

Comparing experimental and theoretical probability

Theoretical probability is what should happen based on math.

Experimental probability is what actually happened in your trials or simulation.

These may not match exactly, especially with a small number of trials. That does not mean the simulation is wrong. Randomness causes results to vary.

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

For example, if the theoretical probability is \(\frac{1}{2}\), a small simulation might give \(\frac{6}{10}\), but a much larger simulation might give \(\frac{49}{100}\) or \(\frac{501}{1000}\).

Common mistakes to avoid

• Using an unfair model: Make sure the simulation matches the real probabilities.
• Too few trials: Very small samples can give misleading results.
• Counting successes incorrectly: Be clear about what the event is.
• Mixing up one trial and many trials: A trial may include several random actions, like flipping 3 coins, not just one action.

Helpful questions to ask yourself

• What real event am I trying to model?
• What random tool best matches the outcomes?
• Are all outcomes represented fairly?
• What counts as a success?
• Did I run enough trials to make a good estimate?

Summary

A simulation is a way to use random outcomes to estimate the probability of an event. It is especially useful for compound events or real-world situations that are harder to calculate directly.

To make a good simulation, choose a random model that matches the real situation, define success clearly, run many trials, and then find the experimental probability using

The more trials you run, the more reliable your estimate usually becomes.