Probability

Probability helps us describe how likely something is to happen.

Some things can never happen, some things might happen, and some things will definitely happen. Probability gives us words and numbers to talk about these chances.

In this lesson, you will learn probability terminology and how to place events on a probability scale from 0 to 1.

The probability scale is a number line that shows chance.

It starts at 0 and ends at 1.

$$0 \hspace{1cm} \frac{1}{2} \hspace{1cm} 1$$

Each number on the scale tells how likely an event is:

• 0 means impossible — it cannot happen.
• 1 means certain — it will happen.
• \(\frac{1}{2}\) means equally likely — it has a 50-50 chance.
• A number closer to 0 means the event is unlikely.
• A number closer to 1 means the event is likely.

We also use special words to describe probability. These words help us talk about chance before we use exact numbers.

• Impossible: no chance
• Unlikely: a small chance
• Equally likely: the same chance as not happening
• Likely: a good chance
• Certain: it must happen

Here is how these words fit on the scale:

$$\text{Impossible} \qquad \text{Unlikely} \qquad \text{Equally likely} \qquad \text{Likely} \qquad \text{Certain}$$

$$0 \qquad \text{between }0\text{ and }\frac{1}{2} \qquad \frac{1}{2} \qquad \text{between }\frac{1}{2}\text{ and }1 \qquad 1$$

Let’s think about what these words mean in real life.

• The sun rising tomorrow is certain.
• Rolling a 9 on a standard number cube with numbers 1 to 6 is impossible.
• Flipping a coin and getting heads is equally likely.
• Picking a red marble from a bag with many red marbles and only a few blue marbles is likely.
• Snow falling in the middle of summer in a very hot place might be unlikely.

Important idea: probability can be written as a word or as a number.

For example:

• Impossible means probability of 0.
• Certain means probability of 1.
• Equally likely means probability of \(\frac{1}{2}\).

Numbers between 0 and 1 tell us more exact chances.

For example:

• \(0.1\) is very unlikely.
• \(0.3\) is unlikely.
• \(0.5\) is equally likely.
• \(0.8\) is likely.
• \(1\) is certain.

You do not always need to use decimals. In 5th Grade, it is fine to think mostly about 0, \(\frac{1}{2}\), and 1, and whether something is closer to 0 or closer to 1.

A helpful way to remember the scale:

1. Ask: Can it happen? If not, it is impossible, so the probability is 0.
2. Ask: Must it happen? If yes, it is certain, so the probability is 1.
3. Ask: Is it a fair 50-50 chance? If yes, it is equally likely, so the probability is \(\frac{1}{2}\).
4. If it is not 0, \(\frac{1}{2}\), or 1, decide whether it is unlikely or likely.

Now let’s look at some worked examples.

Example 1: Coin flip

You flip a fair coin. What is the probability terminology for getting heads?

A fair coin has 2 sides: heads and tails. Each side has the same chance.

Getting heads has the same chance as not getting heads.

So getting heads is equally likely.

The probability is $$\frac{1}{2}$$

Example 2: Number cube

You roll a standard number cube with numbers 1 through 6. What word describes the chance of rolling a 7?

A standard number cube only has the numbers 1, 2, 3, 4, 5, and 6.

There is no 7 on the cube.

So rolling a 7 is impossible.

The probability is $$0$$

Example 3: Colored marbles

A bag has 8 red marbles and 2 blue marbles. You close your eyes and pick 1 marble. Is picking a red marble unlikely, equally likely, likely, impossible, or certain?

There are more red marbles than blue marbles.

Since red marbles make up most of the bag, picking red has a good chance of happening.

So picking a red marble is likely.

It is not certain, because you could still pick a blue marble.

Example 4: A certain event

You choose 1 letter from the word CAT. What is the chance that you pick a letter?

The choices are C, A, and T. All of them are letters.

No matter which one you pick, it will be a letter.

So this event is certain.

The probability is $$1$$

Let’s compare events on the probability scale.

Suppose these are the events:

• Getting a Monday after Sunday
• Rolling a 10 on a 6-sided number cube
• Flipping tails on a fair coin
• Picking a green marble from a bag with 9 green marbles and 1 yellow marble

We can place them on the scale like this:

• Impossible (0): rolling a 10 on a 6-sided number cube
• Equally likely \(\left(\frac{1}{2}\right)\): flipping tails on a fair coin
• Likely: picking a green marble from a bag with 9 green marbles and 1 yellow marble
• Certain (1): getting a Monday after Sunday

This helps us compare which events have smaller or greater chances.

Common mistakes to avoid

• Do not say something is certain just because it is very likely. If there is any chance it will not happen, it is not certain.
• Do not say something is impossible if it is just hard to happen. If it can happen, then it is not impossible.
• Remember that equally likely means a true 50-50 chance.

Quick practice thinking

Try naming each event:

• Picking an even number from the numbers 2, 4, 6, 8
• Rolling a number less than 7 on a standard number cube
• Picking a purple crayon from a box with 1 purple and 9 orange crayons
• Flipping a fair coin and getting heads

Answers:

• Certain — all the numbers are even.
• Certain — every number on a standard number cube is less than 7.
• Unlikely — only 1 out of 10 crayons is purple.
• Equally likely — heads and tails have the same chance.

Summary

Probability tells how likely an event is to happen.

We use the probability scale from 0 to 1:

• 0 = impossible
• between 0 and \(\frac{1}{2}\) = unlikely
• \(\frac{1}{2}\) = equally likely
• between \(\frac{1}{2}\) and 1 = likely
• 1 = certain

When you think about a chance event, ask yourself whether it cannot happen, must happen, has a 50-50 chance, or is somewhere in between. That will help you choose the correct probability word and place it on the scale.

Sample Spaces and Outcomes

When we talk about probability, we are thinking about what might happen. To understand probability well, we first need to know all the possible results of an event.

That is where outcomes and sample spaces come in.

An outcome is one possible result.

A sample space is the full list of all possible outcomes.

If we miss an outcome, our probability answer can be wrong. So it is very important to make a complete list.

Why is this important?

Suppose you flip a coin. If you want to know the chance of getting heads, you need to know all the possible results first. A coin can land on:

• Heads
• Tails

So the sample space is:

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

There are 2 outcomes in this sample space.

Once we know the sample space, we can answer questions about probability more carefully.

Main Idea 1: What is an outcome?

An outcome is just one thing that can happen.

• If you roll a number cube, getting a 4 is one outcome.
• If you spin a spinner, landing on blue is one outcome.
• If you flip a coin, getting tails is one outcome.

Each different result counts as a different outcome.

Main Idea 2: What is a sample space?

The sample space is the set of every possible outcome.

For a number cube with numbers 1 through 6, the sample space is:

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

This sample space has 6 outcomes.

For a spinner with colors red, blue, and green, the sample space is:

$$\{\text{red},\ \text{blue},\ \text{green}\}$$

This sample space has 3 outcomes.

Main Idea 3: The sample space must be complete

A complete sample space includes every possible outcome exactly once in the list.

For example, if you roll a number cube and write:

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

that sample space is not complete because 6 is missing.

If you write:

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

that is also not a good sample space list, because 2 is repeated. We want each possible outcome listed clearly.

Main Idea 4: Ways to show a sample space

There are different ways to organize all possible outcomes. In 5th Grade, common ways are:

• Lists
• Tables
• Arrays

These tools help us stay organized and make sure we do not forget anything.

Method 1: Using a list

A list works well when there are not too many outcomes.

Example: flip a coin once.

Sample space:

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

Example: roll a number cube once.

Sample space:

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

Method 2: Using a table

A table is helpful when an event has two parts, like flipping a coin and rolling a number cube.

One part can label the rows, and the other part can label the columns.

Method 3: Using an array

An array is a neat grid that shows combinations of outcomes. It helps us see patterns and count carefully.

For example, if one coin flip can be H or T, and one spinner can land on 1, 2, or 3, we can arrange the outcomes in rows and columns.

Worked Example 1: One simple event

Question: What is the sample space for rolling a number cube one time?

Step 1: Think of all the numbers on the number cube.

• 1
• 2
• 3
• 4
• 5
• 6

Step 2: Write the sample space.

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

Answer: The sample space is $$\{1,2,3,4,5,6\}$$ and it has 6 outcomes.

Worked Example 2: Two-part event with a list

Question: What is the sample space for flipping a coin and then rolling a number cube?

Step 1: List the coin outcomes.

• H
• T

Step 2: List the number cube outcomes.

• 1, 2, 3, 4, 5, 6

Step 3: Match each coin result with each number result.

The possible outcomes are:

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

Step 4: Count them.

There are 12 outcomes.

Answer: The sample space is $$\{H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6\}$$.

Worked Example 3: Using a table

Question: A spinner has colors red and blue. A number cube is rolled. What is the sample space?

Step 1: Write the spinner outcomes.

• red
• blue

Step 2: Write the number cube outcomes.

• 1, 2, 3, 4, 5, 6

Step 3: Make all combinations.

Step 4: Write the sample space as a list if needed.

$$\{\text{red 1},\text{ red 2},\text{ red 3},\text{ red 4},\text{ red 5},\text{ red 6},\text{ blue 1},\text{ blue 2},\text{ blue 3},\text{ blue 4},\text{ blue 5},\text{ blue 6}\}$$

Answer: There are 12 outcomes in the sample space.

Worked Example 4: Two spinners

Question: Spinner A has outcomes 1 and 2. Spinner B has outcomes X, Y, and Z. What is the sample space?

Step 1: Write the outcomes for each spinner.

• Spinner A: 1, 2
• Spinner B: X, Y, Z

Step 2: Pair each outcome from Spinner A with each outcome from Spinner B.

For 1, the outcomes are:

• 1X
• 1Y
• 1Z

For 2, the outcomes are:

• 2X
• 2Y
• 2Z

Step 3: Write the complete sample space.

$$\{1X,1Y,1Z,2X,2Y,2Z\}$$

Answer: The sample space has 6 outcomes.

How to make sure your sample space is complete

Use these questions to check your work:

• Did I list all the possible outcomes?
• Did I leave anything out?
• Did I repeat any outcome by mistake?
• Did I organize my work so it is easy to see?

Helpful strategy

When there are two parts to an event, keep one part fixed while changing the other part.

For example, with a coin and a number cube:

• Start with H: H1, H2, H3, H4, H5, H6
• Then use T: T1, T2, T3, T4, T5, T6

This helps you avoid missing outcomes.

Common mistakes

• Forgetting outcomes — not making a complete list.
• Repeating outcomes — writing the same result more than once.
• Mixing up order — if the event happens in order, the order matters.

For example, if you flip a coin and then roll a number cube, writing H3 means something different from 3H because the event was described in a certain order.

Connecting sample spaces to probability

Once we know the sample space, we can count how many outcomes are possible.

If all outcomes are equally likely, probability can be found using:

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

So building the sample space is the first important step.

Quick Practice Questions

1. What is the sample space for flipping a coin once?
2. What is the sample space for a spinner with outcomes A, B, and C?
3. What is the sample space for rolling a number cube and flipping a coin?
4. A spinner has red and yellow. Another spinner has 1, 2, 3, 4. How many outcomes are in the sample space?

Answers to Quick Practice

1. $$\{H,T\}$$
2. $$\{A,B,C\}$$
3. $$\{1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T\}$$ or another correct ordered list, depending on how the event is written
4. There are 8 outcomes because $$2 \times 4 = 8$$

Summary

An outcome is one possible result of an event.

A sample space is the complete list of all possible outcomes.

You can show a sample space with a list, table, or array.

When making a sample space, be sure to include every possible outcome, do not repeat outcomes, and stay organized. This helps you understand probability and solve problems correctly.

Theoretical vs. Experimental Probability

Probability helps us describe how likely something is to happen.

Sometimes we can figure out the probability by thinking about all the possible outcomes. Other times we test it by doing an experiment many times and looking at what actually happens.

These are called theoretical probability and experimental probability. In this lesson, you will learn what each one means, how to find them, and how they are alike and different.

1. What is probability?

Probability is the chance that an event will happen.

• An event is something that might happen, like rolling a 4 on a number cube.
• A probability can be written as a fraction, decimal, or percent.
• Probabilities go from 0 to 1.

If an event is impossible, its probability is 0.

If an event is certain, its probability is 1.

For example:

• The probability of rolling a 7 on a standard number cube is 0 because it is impossible.
• The probability of rolling a number less than 7 on a standard number cube is 1 because it is certain.

2. Theoretical probability

Theoretical probability is what we expect to happen based on math.

We find it by comparing:

• the number of favorable outcomes (the outcomes we want), and
• the number of total possible outcomes.

The formula is:

This works best when all outcomes are equally likely, like flipping a fair coin or rolling a fair number cube.

Example 1: Theoretical probability with a coin

You flip a fair coin. What is the theoretical probability of landing on heads?

There are 2 possible outcomes: heads and tails.

There is 1 favorable outcome: heads.

So the theoretical probability of heads is \(\frac{1}{2}\).

Example 2: Theoretical probability with a number cube

You roll a fair 6-sided number cube. What is the theoretical probability of rolling an even number?

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

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

So the theoretical probability of rolling an even number is \(\frac{1}{2}\).

3. Experimental probability

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

We find it by comparing:

• the number of times the event happened, and
• the total number of trials.

A trial is one test, like one coin flip or one roll of a number cube.

The formula is:

Example 3: Experimental probability with a coin

A coin is flipped 10 times. It lands on heads 6 times. What is the experimental probability of heads?

Heads happened 6 times.

There were 10 total flips.

So the experimental probability of heads is \(\frac{6}{10}\), or \(\frac{3}{5}\).

Notice that this is not exactly the same as the theoretical probability of \(\frac{1}{2}\). That can happen, especially when there are only a few trials.

4. How are they different?

Theoretical probability is based on what should happen.

Experimental probability is based on what did happen.

• Theoretical probability: uses reasoning and all possible outcomes.
• Experimental probability: uses data from actual trials.

Here is a simple comparison:

• If you ask, “What is the chance of pulling a red marble from a bag with 3 red and 2 blue marbles?” you are finding theoretical probability.
• If you actually pull a marble, put it back, and do this 20 times, then use the results, you are finding experimental probability.

5. How are they alike?

Both theoretical and experimental probability tell us about chance.

• Both can be written as fractions.
• Both compare part of the outcomes to the whole.
• Both help us predict what may happen.

6. Experimental results can be close to theoretical results

When an experiment is repeated many times, the experimental probability often gets closer to the theoretical probability.

For example, the theoretical probability of rolling a 6 on a fair number cube is:

If you roll only 6 times, you might get a 6 zero times, one time, or even two times. But if you roll 600 times, the experimental probability will often be much closer to \(\frac{1}{6}\).

This does not mean the results will be perfect. It means that more trials usually give a better picture of the true chance.

Example 4: Comparing theoretical and experimental probability

A bag has 4 green cubes and 1 yellow cube. You reach in without looking, choose 1 cube, record the color, and put it back. You do this 20 times. You get green 14 times.

Step 1: Find the theoretical probability of green.

There are 5 cubes total, and 4 are green.

Step 2: Find the experimental probability of green.

Green happened 14 times out of 20 trials.

Step 3: Compare them.

• Theoretical probability: \(\frac{4}{5}\)
• Experimental probability: \(\frac{7}{10}\)

These are not equal, but they are both reasonable. If the experiment were repeated many more times, the experimental probability might move closer to \(\frac{4}{5}\).

7. Steps for solving probability problems

When you solve a probability problem, ask yourself:

1. Am I using math to predict, or am I using results from an experiment?
2. If it is theoretical probability, how many outcomes are possible?
3. If it is theoretical probability, how many outcomes are favorable?
4. If it is experimental probability, how many times did the event happen?
5. If it is experimental probability, how many total trials were there?
6. Can I simplify the fraction?

8. Common mistakes to avoid

• Mixing up the two types: Theoretical uses possible outcomes. Experimental uses actual results.
• Using the wrong total: For theoretical probability, use total possible outcomes. For experimental probability, use total trials.
• Forgetting to simplify: For example, \(\frac{3}{6}\) should be simplified to \(\frac{1}{2}\).
• Thinking they must always match: They do not always match exactly, especially with only a small number of trials.

9. Quick practice thinking

Decide whether each one is theoretical or experimental probability:

• “What is the chance of choosing a vowel from the letters A, B, C, D, E?” → Theoretical
• “A spinner was spun 30 times and landed on blue 8 times. What is the probability of blue based on the results?” → Experimental

Summary

Theoretical probability is the probability you calculate using all possible outcomes.

Experimental probability is the probability you find using actual results from trials.

Theoretical probability tells what should happen. Experimental probability tells what did happen. When there are more and more trials, the experimental probability often gets closer to the theoretical probability.

Expressing Probability means showing how likely something is to happen using numbers. In 5th Grade, we often write probability as a fraction, a decimal, and a percentage.

Probability helps us answer questions like: What is the chance of flipping heads? What is the chance of picking a red marble? What is the chance of rolling an even number?

A probability tells us how likely an event is. It is always 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 might happen.

We can also think of these as percentages:

• 0% means impossible.
• 100% means certain.
• Anything between 0% and 100% shows some chance of happening.

The basic probability rule is:

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

Favorable outcomes are the outcomes we want. Total possible outcomes means all outcomes that can happen.

For example, if you roll a fair number cube with 6 sides and want an even number, the even numbers are 2, 4, and 6. That gives 3 favorable outcomes out of 6 total outcomes, so the probability is:

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

That same probability can also be written as:

• Fraction: \(\frac{1}{2}\)
• Decimal: \(0.5\)
• Percentage: \(50\%\)

Why do we use fractions, decimals, and percentages? They all show the same chance, just in different forms. Sometimes one form is easier to understand or compare than another.

Here are some common probability values:

• \(\frac{1}{4} = 0.25 = 25\%\)
• \(\frac{1}{2} = 0.5 = 50\%\)
• \(\frac{3}{4} = 0.75 = 75\%\)
• \(1 = 1.0 = 100\%\)

How to express probability in different ways

1. Write the probability as a fraction.
2. Simplify the fraction if possible.
3. Change the fraction to a decimal by dividing the numerator by the denominator.
4. Change the decimal to a percentage by multiplying by 100, or moving the decimal point 2 places to the right.

Let’s work through some examples.

Worked Example 1: Picking a red marble

A bag has 5 marbles: 2 red and 3 blue. What is the probability of picking a red marble?

Step 1: Count favorable outcomes. There are 2 red marbles, so there are 2 favorable outcomes.

Step 2: Count total possible outcomes. There are 5 marbles in all.

$$P(\text{red}) = \frac{2}{5}$$

Step 3: Write as a decimal.

$$2 \div 5 = 0.4$$

Step 4: Write as a percentage.

$$0.4 = 40\%$$

Answer:

• Fraction: \(\frac{2}{5}\)
• Decimal: \(0.4\)
• Percentage: \(40\%\)

Worked Example 2: Flipping a coin

A fair coin has 2 possible outcomes: heads or tails. What is the probability of flipping heads?

There is 1 favorable outcome and 2 total outcomes.

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

Now change it to decimal and percentage:

$$1 \div 2 = 0.5$$

$$0.5 = 50\%$$

Answer:

• Fraction: \(\frac{1}{2}\)
• Decimal: \(0.5\)
• Percentage: \(50\%\)

Worked Example 3: Rolling a number greater than 4

You roll a fair 6-sided number cube. What is the probability of rolling a number greater than 4?

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

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

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

Simplify the fraction:

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

Now write it as a decimal:

$$1 \div 3 = 0.333\ldots$$

This decimal repeats, so we can write it as about \(0.33\).

Now write it as a percentage:

$$0.333\ldots = 33.3\ldots\%$$

We can say it is about \(33\%\).

Answer:

• Fraction: \(\frac{1}{3}\)
• Decimal: about \(0.33\)
• Percentage: about \(33\%\)

Worked Example 4: Choosing a letter

The letters in the word APPLE are written on cards. One card is picked at random. What is the probability of picking the letter P?

The word APPLE has 5 letters: A, P, P, L, E.

There are 2 cards with P, so there are 2 favorable outcomes out of 5 total outcomes.

$$P(\text{P}) = \frac{2}{5}$$

Change to decimal and percentage:

$$2 \div 5 = 0.4$$

$$0.4 = 40\%$$

Answer:

• Fraction: \(\frac{2}{5}\)
• Decimal: \(0.4\)
• Percentage: \(40\%\)

Important things to remember

• The numerator tells how many outcomes you want.
• The denominator tells how many total outcomes there are.
• The probability of an event cannot be less than 0 or more than 1.
• A fraction, decimal, and percentage can all name the same probability.
• Simplify fractions when you can.

Common mistakes to avoid

• Forgetting the total number of outcomes. Always count all possible outcomes.
• Using the wrong favorable outcomes. Make sure you count only the outcomes that match the event.
• Not simplifying the fraction. For example, \(\frac{3}{6}\) should be simplified to \(\frac{1}{2}\).
• Mixing up decimals and percentages. For example, \(0.5\) means \(50\%\), not \(5\%\).

Quick check

Suppose a spinner has 8 equal sections, and 6 of them are green.

• Favorable outcomes: 6
• Total outcomes: 8

$$P(\text{green}) = \frac{6}{8} = \frac{3}{4}$$

As a decimal:

$$3 \div 4 = 0.75$$

As a percentage:

$$0.75 = 75\%$$

So the probability of landing on green is \(\frac{3}{4}\), \(0.75\), or \(75\%\).

Summary

To express probability, first write the number of favorable outcomes over the total number of possible outcomes. Then you can change that fraction into a decimal and a percentage.

Remember:

• Fraction: shows part of the whole
• Decimal: shows the same amount in decimal form
• Percentage: shows the amount out of 100

All three forms help us describe chance clearly and exactly.

Compound Events and Tree Diagrams

Sometimes in probability, we look at just one event, like flipping one coin or rolling one die. But many questions ask about more than one event. For example, what happens when you flip two coins, or choose a shirt and then a hat?

These are called compound events. A compound event happens when there are two or more chances or steps.

A very helpful way to organize all possible outcomes of a compound event is with a tree diagram. A tree diagram shows each choice as a branch, so we can see every possible result clearly.

In this lesson, you will learn how to:

• understand what a compound event is,
• draw and read a tree diagram,
• list all possible outcomes,
• and use the outcomes to find probability.

1. What is a compound event?

A compound event is an event made of two or more simple events.

Here are some examples:

• flipping a coin and then flipping it again,
• rolling a die and then rolling it again,
• choosing a snack and then choosing a drink,
• spinning a spinner and flipping a coin.

Each step has its own possible outcomes. When we put the steps together, we get a larger set of possible outcomes.

2. What is a tree diagram?

A tree diagram is a picture that uses branches to show all possible outcomes in order.

It starts with the first event. Then each branch splits again for the second event. If there are more events, the branches keep splitting.

Each path from start to finish shows one complete outcome.

For example, if you flip a coin two times, the first flip can be:

• H for heads
• T for tails

Then from each of those, the second flip can also be:

• H
• T

So the complete outcomes are:

• HH
• HT
• TH
• TT

There are 4 possible outcomes.

3. How to make a tree diagram

Follow these steps:

1. Write the possible outcomes for the first event.
2. From each of those, draw branches for the possible outcomes of the second event.
3. If there is a third event, keep branching again.
4. List the outcomes at the ends of the branches.
5. Count the total number of outcomes.

Be careful to make sure every branch is included. Missing a branch means missing an outcome.

4. Using a tree diagram to find probability

Once we know all possible outcomes, we can find probability with this idea:

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

Favorable outcomes are the outcomes that match the event we want.

For example, if there are 4 total outcomes and 2 of them match the event, then:

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

Worked Example 1: Flipping two coins

Question: If you flip a coin two times, what is the probability of getting 2 heads?

Step 1: List the outcomes.

The first flip can be H or T. The second flip can be H or T.

So the possible outcomes are:

• HH
• HT
• TH
• TT

Step 2: Count the total outcomes.

There are 4 total outcomes.

Step 3: Count the favorable outcomes.

Only HH has 2 heads, so there is 1 favorable outcome.

Step 4: Write the probability.

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

So, the probability of getting 2 heads is \(\frac{1}{4}\).

Worked Example 2: Rolling a die and flipping a coin

Question: You roll a number cube with numbers 1 through 6 and then flip a coin. What is the probability of rolling an even number and then getting heads?

Step 1: First event outcomes.

The die can land on 1, 2, 3, 4, 5, or 6.

Step 2: Second event outcomes.

For each number, the coin can be H or T.

Step 3: Count all possible outcomes.

Each of the 6 numbers can go with 2 coin results, so there are:

$$6 \times 2 = 12$$

So there are 12 total outcomes.

Step 4: Find the favorable outcomes.

Even numbers are 2, 4, and 6.

We want heads, so the favorable outcomes are:

• 2H
• 4H
• 6H

There are 3 favorable outcomes.

Step 5: Write the probability.

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

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

Worked Example 3: Choosing an outfit

Question: Mia chooses 1 shirt and 1 pair of pants. She has 2 shirts: red and blue. She has 3 pairs of pants: black, gray, and tan. How many different outfits can she make?

Step 1: Start with the shirts.

• red
• blue

Step 2: From each shirt, branch to the pants.

• black
• gray
• tan

Step 3: List the full outcomes.

• red-black
• red-gray
• red-tan
• blue-black
• blue-gray
• blue-tan

There are 6 different outfits.

We can also see this with multiplication:

$$2 \times 3 = 6$$

The tree diagram helps us make sure we did not miss any outfit.

Worked Example 4: Spinning and flipping

Question: A spinner has 3 equal sections: red, yellow, and green. Then a coin is flipped. What is the probability of landing on green and then getting tails?

Step 1: List all possible outcomes.

• red-heads
• red-tails
• yellow-heads
• yellow-tails
• green-heads
• green-tails

There are 6 total outcomes.

Step 2: Find the favorable outcomes.

We want green and tails. That is only:

• green-tails

So there is 1 favorable outcome.

Step 3: Write the probability.

$$\frac{1}{6}$$

So, the probability is \(\frac{1}{6}\).

5. Important ideas to remember

• A compound event has more than one step.
• A tree diagram shows all possible outcomes using branches.
• Each complete path in the tree is one outcome.
• To find probability, count favorable outcomes and divide by total outcomes.
• Tree diagrams help you stay organized and avoid missing outcomes.

6. Common mistakes to avoid

• Forgetting outcomes: Make sure every branch splits fully.
• Not keeping the order: In some compound events, order matters. For example, HT and TH are different outcomes when flipping two coins.
• Using the wrong total: Always count all the possible outcomes before finding probability.
• Counting only one step: Remember that a compound event includes all steps together.

7. Quick practice questions

Try these on your own:

1. If you flip a coin twice, what is the probability of getting exactly 1 head?
2. If you roll a die and flip a coin, how many total outcomes are there?
3. You choose 1 sandwich from turkey or cheese, and 1 drink from milk, juice, or water. How many meal combinations are possible?

Answers:

1. The outcomes are HH, HT, TH, TT. Exactly 1 head happens in HT and TH, so the probability is \(\frac{2}{4} = \frac{1}{2}\).
2. There are \(6 \times 2 = 12\) total outcomes.
3. There are \(2 \times 3 = 6\) meal combinations.

Summary

A compound event has two or more parts. A tree diagram helps us list all possible outcomes in an organized way. After listing the outcomes, we can find probability by dividing the number of favorable outcomes by the total number of outcomes.

When you solve these problems, go step by step, check every branch, and count carefully. Tree diagrams are a great tool for understanding probability.