Probability and Chance

Probability is a way to describe how likely something is to happen.

We use probability every day, even if we do not say the word. For example, you might say, “It will probably rain,” or “I am sure the bus will come soon.” In math, probability helps us describe these chances using numbers.

The probability of an event is shown on a scale from 0 to 1.

• 0 means the event is impossible. It cannot happen.
• 1 means the event is certain. It will definitely happen.
• A number between 0 and 1 means the event is possible, but not guaranteed.

You can think of probability like a number line:

Impossible  Possible  Certain

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

If an event has a probability close to 0, it is unlikely. If it has a probability close to 1, it is likely. If it is right in the middle, around \(\frac{1}{2}\), it is equally likely to happen or not happen.

Ways to describe likelihood

We can describe chance with words or numbers.

• Impossible  probability \(0\)
• Unlikely  probability closer to \(0\)
• Equally likely  probability \(\frac{1}{2}\) or \(0.5\) or \(50\%\)
• Likely  probability closer to \(1\)
• Certain  probability \(1\)

This means probability can be written in different forms:

• as a fraction, like \(\frac{1}{4}\)
• as a decimal, like \(0.25\)
• as a percent, like \(25\%\)

These forms all mean the same amount. For example:

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

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

$$\frac{3}{4} = 0.75 = 75\%$$

Finding probability

To find the probability of an event, use this rule:

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

A favorable outcome is an outcome you want to happen.

The total number of possible outcomes means all the different results that could happen.

This rule works best when all outcomes are equally likely. That means each outcome has the same chance.

For example, when rolling a fair number cube, the numbers 1 through 6 are all equally likely.

Worked Example 1: Rolling a number cube

Question: What is the probability of rolling a 3 on a fair 6-sided number cube?

Step 1: Count the favorable outcomes.

There is only 1 favorable outcome: rolling a 3.

Step 2: Count the total possible outcomes.

A 6-sided number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6.

Step 3: Write the probability.

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

Step 4: Write it in decimal and percent form.

$$\frac{1}{6} \approx 0.17 \approx 17\%$$

Answer: The probability of rolling a 3 is \(\frac{1}{6}\), about \(0.17\), or about \(17\%\).

This is unlikely because it is closer to 0 than to 1.

Worked Example 2: Picking a colored marble

Question: A bag has 4 red marbles and 6 blue marbles. What is the probability of picking a red marble?

Step 1: Count the favorable outcomes.

There are 4 red marbles.

Step 2: Count the total possible outcomes.

There are \(4 + 6 = 10\) marbles altogether.

Step 3: Write the probability.

$$\frac{4}{10}$$

Step 4: Simplify if possible.

$$\frac{4}{10} = \frac{2}{5}$$

Step 5: Change to decimal and percent.

$$\frac{2}{5} = 0.4 = 40\%$$

Answer: The probability of picking a red marble is \(\frac{2}{5}\), or \(0.4\), or \(40\%\).

This is still unlikely, but it is closer to equally likely than Example 1.

Worked Example 3: Probability of an even number

Question: What is the probability of rolling an even number on a fair 6-sided number cube?

Step 1: List the even numbers.

The even numbers are 2, 4, and 6.

Step 2: Count the favorable outcomes.

There are 3 favorable outcomes.

Step 3: Count the total possible outcomes.

There are 6 total possible outcomes.

Step 4: Write the probability.

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

Step 5: Change to decimal and percent.

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

Answer: The probability of rolling an even number is \(\frac{1}{2}\), or \(0.5\), or \(50\%\).

This means rolling an even number is equally likely as not rolling an even number.

Worked Example 4: A spinner

Question: A spinner has 8 equal sections. 6 sections are green and 2 sections are yellow. What is the probability of landing on green?

Step 1: Count the favorable outcomes.

There are 6 green sections.

Step 2: Count the total possible outcomes.

There are 8 sections in all.

Step 3: Write the probability.

$$\frac{6}{8}$$

Step 4: Simplify.

$$\frac{6}{8} = \frac{3}{4}$$

Step 5: Change to decimal and percent.

$$\frac{3}{4} = 0.75 = 75\%$$

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

This is likely because it is close to 1.

Special probability values

Some probabilities are important to recognize quickly.

• If an event cannot happen, its probability is \(0\).
• If an event must happen, its probability is \(1\).
• If an event has the same chance of happening as not happening, its probability is \(\frac{1}{2}\).

Examples:

• Rolling a 7 on a 6-sided number cube: \(0\) because it is impossible.
• Picking a month from the year and getting a month with fewer than 32 days: \(1\) because every month has fewer than 32 days.
• Flipping a fair coin and getting heads: \(\frac{1}{2}\) because heads and tails are equally likely.

Comparing probabilities

You can compare probabilities to decide which event is more likely.

A larger probability means a greater chance of happening.

For example:

• \(0.8\) is greater than \(0.3\), so an event with probability \(0.8\) is more likely.
• \(25\%\) is less than \(50\%\), so an event with probability \(25\%\) is less likely.
• \(\frac{3}{4}\) is greater than \(\frac{1}{4}\), so \(\frac{3}{4}\) is more likely.

Sometimes it helps to write them in the same form before comparing.

For example:

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

$$\frac{3}{10} = 0.3 = 30\%$$

Since \(50\% > 30\%\), \(\frac{1}{2}\) is more likely than \(\frac{3}{10}\).

Using words, fractions, decimals, and percents together

It is important to move between different ways of showing probability.

Common mistakes to avoid

• Do not make the probability bigger than 1. Probability must be from 0 to 1.
• Do not forget the total number of outcomes. The denominator shows all possible outcomes.
• Do not confuse favorable outcomes with total outcomes. Favorable outcomes are only the ones you want.
• Remember to simplify fractions when possible.

For example, if a bag has 3 red marbles and 5 blue marbles, the probability of red is not \(\frac{3}{5}\). It is:

$$\frac{3}{8}$$

That is because there are 8 marbles in total, not 5.

Quick practice ideas

Try asking yourself these questions:

1. If you flip a fair coin, what is the probability of tails?
2. If a bag has 7 white beads and 3 black beads, what is the probability of black?
3. If a spinner has 5 equal sections and all 5 are orange, what is the probability of orange?
4. If you roll a fair number cube, what is the probability of rolling a number greater than 4?

You can solve each one by using:

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

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.

You can write probability as a fraction, decimal, or percent.

To find probability, divide the number of favorable outcomes by the total number of possible outcomes.

Then decide whether the event is impossible, unlikely, equally likely, likely, or certain.

Experimental Probability and Relative Frequency

Sometimes we want to know how likely something is to happen. In math, this is called probability.

One way to find probability is by using actual results from an experiment. When we use results we observed, we are finding experimental probability.

Another name for this idea is relative frequency. Relative frequency tells us how often an event happened compared to the total number of trials.

This lesson will show you how to calculate experimental probability and relative frequency, how to write the answers as fractions, decimals, or percents, and how to use data from real experiments.

1. What is an experiment?

An experiment is an action or trial that has a result you can record.

• Flipping a coin
• Rolling a number cube
• Spinning a spinner
• Pulling a colored cube from a bag

Each time you do the action, it is called a trial.

For example, if you flip a coin 20 times, you have done 20 trials.

2. What is experimental probability?

Experimental probability is the probability of an event based on what actually happened in an experiment.

The formula is:

This means:

• The top number tells how many times the event happened.
• The bottom number tells how many trials there were in all.

3. What is relative frequency?

Relative frequency is another way to describe the same idea. It compares the number of times an event happened to the total number of trials.

So, in this lesson, experimental probability and relative frequency are found the same way:

For example, if a spinner lands on red 8 times out of 20 spins, then:

So the experimental probability, or relative frequency, of landing on red is 2/5, or 0.4, or 40%.

4. Important words to know

• Outcome: a possible result, like heads on a coin toss
• Event: the result you are interested in, like rolling an even number
• Trial: one time doing the experiment
• Frequency: the number of times something happened
• Total trials: the number of times the experiment was done

5. How to find experimental probability

Follow these steps:

1. Identify the event you want to study.
2. Count how many times the event happened.
3. Count the total number of trials.
4. Write a fraction: event happened over total trials.
5. Simplify if possible.
6. If needed, change the answer to a decimal or percent.

6. Worked Example 1: Flipping a coin

A coin is flipped 12 times. It lands on heads 7 times. What is the experimental probability of heads?

Step 1: Number of times heads happened = 7

Step 2: Total number of flips = 12

Use the formula:

This fraction cannot be simplified.

As a decimal,

As a percent, that is about 58%.

Answer: The experimental probability of heads is \(\frac{7}{12}\), or about 0.58, or about 58%.

7. Worked Example 2: Rolling a number cube

A number cube is rolled 30 times. The number 4 appears 6 times. What is the relative frequency of rolling a 4?

Step 1: Event happened 6 times

Step 2: Total trials = 30

Simplify:

As a decimal:

As a percent:

Answer: The relative frequency of rolling a 4 is \(\frac{1}{5}\), or 0.2, or 20%.

8. Worked Example 3: Using a table of data

A spinner was spun 25 times. The results are shown below:

• Red: 9
• Blue: 7
• Green: 5
• Yellow: 4

What is the experimental probability of landing on blue?

Step 1: Blue happened 7 times.

Step 2: Total spins = 25.

This fraction is already simplified.

As a decimal:

As a percent:

Answer: The experimental probability of blue is \(\frac{7}{25}\), or 0.28, or 28%.

9. Worked Example 4: Finding probability of a group of outcomes

A bag contains tiles with numbers. A tile is chosen, recorded, and put back. This is done 40 times. The results are:

• 1: 8 times
• 2: 11 times
• 3: 9 times
• 4: 12 times

What is the experimental probability of choosing an even number?

Even numbers here are 2 and 4.

Step 1: Count how many times an even number happened.

Step 2: Total trials = 40

As a decimal:

As a percent:

Answer: The experimental probability of choosing an even number is \(\frac{23}{40}\), or 0.575, or 57.5%.

10. Writing answers in different forms

Experimental probability can be written in three common forms:

• Fraction: \(\frac{3}{10}\)
• Decimal: \(0.3\)
• Percent: \(30\%\)

These forms all mean the same amount. Be ready to change from one form to another.

11. Experimental probability compared to what we expect

Sometimes the results of an experiment are not exactly what we expect.

For example, if you flip a coin 10 times, you might not get exactly 5 heads and 5 tails. You might get 6 heads and 4 tails, or 7 heads and 3 tails.

This is normal. In a small number of trials, results can vary.

If you do more trials, the experimental probability often gets closer to the probability you would expect.

For example, if you flip a coin many, many times, the relative frequency of heads will often get closer to \(\frac{1}{2}\), or 50%.

12. Why experimental probability is useful

Experimental probability is useful because it is based on real data.

It helps us:

• study what actually happened
• make predictions from data
• compare different results
• understand chance in real-life situations

For example, if a basketball player makes 18 out of 30 free throws in practice, the experimental probability of making a free throw is:

This can help predict how often the player may score on future free throws.

13. Common mistakes to avoid

• Using the wrong total: Always use the total number of trials in the denominator.
• Forgetting to add results: If the event includes more than one outcome, add those frequencies first.
• Not simplifying fractions: Simplify when possible.
• Mixing up event and trials: The numerator is how many times the event happened, not how many times it could have happened.

14. Quick check

Suppose a spinner is spun 15 times and lands on green 3 times.

The experimental probability of green is:

So green happened in 20% of the spins.

15. Summary

Experimental probability tells how likely an event is based on actual results from an experiment.

Relative frequency is another name for comparing how often an event happened to the total number of trials.

Use this formula:

You can write the answer as a fraction, decimal, or percent.

The more trials you do, the more the experimental probability may settle into a more consistent pattern.

Lesson: Theoretical Probability of Simple Events

Probability helps us describe how likely something is to happen. We use probability when we talk about chance, such as getting heads on a coin flip or rolling an even number on a die.

In this lesson, you will learn about theoretical probability. This means figuring out probability using math, not by doing the experiment over and over.

A simple event is an event with one outcome or a small group of outcomes from one action. For example, rolling a 3 on a number cube is a simple event. Getting an even number on one roll is also a simple event because it comes from one roll.

1. What is theoretical probability?

Theoretical probability tells how likely an event is to happen based on all the possible outcomes, when each outcome is equally likely.

For example, on a fair 6-sided number cube, each number from 1 to 6 has the same chance of being rolled.

The formula is:

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

Favorable outcomes are the outcomes you want. Total possible outcomes means all outcomes that could happen.

2. Understanding the sample space

The sample space is the list of all possible outcomes.

Here are some sample spaces:

• For flipping a coin: \(\{\text{H, T}\}\)
• For rolling a 6-sided number cube: \(\{1,2,3,4,5,6\}\)
• For spinning a spinner with 4 equal sections labeled A, B, C, D: \(\{A,B,C,D\}\)

To find theoretical probability, first identify the sample space. Then count how many outcomes are favorable.

3. Probability as a fraction, decimal, or percent

Probabilities are usually written as fractions, but they can also be written as decimals or percents.

For example, if the probability is \(\frac{1}{2}\), it can also be written as:

• Decimal: \(0.5\)
• Percent: \(50\%\)

4. The size of probability

A probability must always be between 0 and 1.

• \(0\) means the event is impossible.
• \(1\) means the event is certain.
• A probability closer to 1 means the event is more likely.
• A probability closer to 0 means the event is less likely.

Examples:

• The probability of rolling a 7 on a standard 6-sided number cube is \(0\).
• The probability of rolling a number less than 7 on a standard 6-sided number cube is \(1\).

5. Steps for finding theoretical probability

1. List or think about all possible outcomes.
2. Count the total number of outcomes.
3. Count the favorable outcomes.
4. Write the probability as a fraction:

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

Then simplify the fraction if possible.

Worked Example 1: Flipping a coin

Question: What is the theoretical probability of flipping heads on a fair coin?

Step 1: List the sample space: \(\{\text{H, T}\}\)

Step 2: Count total outcomes: 2

Step 3: Count favorable outcomes: 1 outcome is heads

Step 4: Write the probability:

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

So, the probability of flipping heads is \(\frac{1}{2}\), or \(0.5\), or \(50\%\).

Worked Example 2: Rolling a number cube

Question: What is the theoretical probability of rolling an even number on a fair 6-sided number cube?

Step 1: List the sample space: \(\{1,2,3,4,5,6\}\)

Step 2: Find the even numbers: \(2,4,6\)

Step 3: Count outcomes:

• Favorable outcomes: 3
• Total outcomes: 6

Step 4: Write the probability:

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

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

Worked Example 3: Spinner

A spinner has 8 equal sections labeled: 1, 2, 3, 4, 5, 6, 7, 8.

Question: What is the theoretical probability of landing on a number greater than 5?

Step 1: List the favorable outcomes: numbers greater than 5 are \(6,7,8\)

Step 2: Count outcomes:

• Favorable outcomes: 3
• Total outcomes: 8

Step 3: Write the probability:

$$P(\text{number greater than 5}) = \frac{3}{8}$$

This probability can also be written as \(0.375\) or \(37.5\%\).

Worked Example 4: Bag of colored marbles

A bag has 4 red marbles, 3 blue marbles, and 1 green marble. One marble is chosen without looking.

Question: What is the theoretical probability of choosing a blue marble?

Step 1: Count all marbles:

$$4 + 3 + 1 = 8$$

Step 2: Count favorable outcomes: there are 3 blue marbles

Step 3: Write the probability:

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

So, the probability of choosing a blue marble is \(\frac{3}{8}\).

6. Important idea: equally likely outcomes

Theoretical probability works best when all outcomes are equally likely.

If a coin is fair, heads and tails are equally likely. If a number cube is fair, each number from 1 to 6 is equally likely. If the sections on a spinner are equal in size, each section is equally likely.

If the outcomes are not equally likely, then this simple probability model does not work the same way.

7. Comparing probabilities

You can compare probabilities to decide which event is more likely.

Example: On a fair 6-sided number cube:

• Probability of rolling a 1 is \(\frac{1}{6}\)
• Probability of rolling a number greater than 3 is \(\frac{3}{6} = \frac{1}{2}\)

Since \(\frac{1}{2} > \frac{1}{6}\), rolling a number greater than 3 is more likely than rolling a 1.

8. Common mistakes to avoid

• Forgetting to count all possible outcomes. Always find the total first.
• Counting the wrong favorable outcomes. Make sure the outcomes match the event in the question.
• Not simplifying the fraction. For example, \(\frac{3}{6}\) should be simplified to \(\frac{1}{2}\).
• Using theoretical probability when outcomes are not equally likely. Check that the object is fair or the sections are equal.

9. Quick practice thinking

Try these in your head:

• What is the probability of rolling a 5 on a fair 6-sided number cube? \(\frac{1}{6}\)
• What is the probability of choosing a vowel from the letters \(\{A, B, C, E\}\)? Favorable outcomes are A and E, so \(\frac{2}{4} = \frac{1}{2}\)
• What is the probability of landing on red if a spinner has 5 equal sections and 2 are red? \(\frac{2}{5}\)

Summary

Theoretical probability tells how likely an event is by comparing the number of favorable outcomes to the total number of equally likely outcomes.

Use this formula:

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

Remember to find the sample space, count carefully, and simplify when you can. Probability helps us make predictions about what might happen before the event takes place.

Sample Spaces for Compound Events

In probability, a sample space is the complete list of all possible outcomes of an event.

A compound event happens when there are two or more actions together, usually in order. For example, flipping a coin and then rolling a number cube is a compound event.

When we find the sample space for a compound event, we must make sure we list every possible outcome and do not leave any out.

This lesson will show you how to make organized lists and tables so you can find all outcomes for compound events.

1. What is a sample space?

A sample space is the set of all outcomes that can happen.

• If you flip one coin, the sample space is: \(\{H, T\}\)
• If you roll one number cube, the sample space is: \(\{1,2,3,4,5,6\}\)

These are simple events because only one action happens.

2. What is a compound event?

A compound event combines two simple events.

• flip a coin and roll a number cube
• spin a spinner and pick a card
• roll one number cube twice

For compound events, each outcome has two parts or more. Order often matters because the actions happen one after another.

For example, rolling a 2 and then a 5 is different from rolling a 5 and then a 2.

3. How to find a sample space for a compound event

There are two very helpful methods:

• organized lists
• tables

Both methods help you make sure the sample space is exhaustive, which means it includes all possible outcomes.

Method A: Organized List

Choose one outcome from the first event, then pair it with every outcome from the second event. Repeat until all first-event outcomes are used.

Method B: Table

Write the outcomes of the first event across the top. Write the outcomes of the second event down the side. Then fill in each box with the matching pair.

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

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

The coin outcomes are \(H\) and \(T\).

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

Use an organized list:

Start with \(H\):

• \((H,1)\)
• \((H,2)\)
• \((H,3)\)
• \((H,4)\)
• \((H,5)\)
• \((H,6)\)

Now use \(T\):

• \((T,1)\)
• \((T,2)\)
• \((T,3)\)
• \((T,4)\)
• \((T,5)\)
• \((T,6)\)

So the sample space is:

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

There are \(2 \times 6 = 12\) total outcomes.

5. Worked Example 2: Roll a number cube twice

Question: What is the sample space for rolling a number cube two times?

The first roll can be \(1,2,3,4,5,6\).

The second roll can also be \(1,2,3,4,5,6\).

We write each outcome as an ordered pair: \((\text{first roll}, \text{second roll})\).

Use an organized list:

When the first roll is 1:

\((1,1),(1,2),(1,3),(1,4),(1,5),(1,6)\)

When the first roll is 2:

\((2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\)

Continue this pattern until the first roll is 6.

The full sample space is:

$$\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),$$

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

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

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

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

$$ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}$$

There are \(6 \times 6 = 36\) outcomes.

Important: \((2,5)\) and \((5,2)\) are different outcomes because the rolls happen in order.

6. Worked Example 3: Use a table for a compound event

Question: A spinner has colors red, blue, and green. Then a coin is flipped. What is the sample space?

Spinner outcomes: \(R, B, G\)

Coin outcomes: \(H, T\)

We can organize the outcomes in a table idea:

• For red: \((R,H), (R,T)\)
• For blue: \((B,H), (B,T)\)
• For green: \((G,H), (G,T)\)

So the sample space is:

$$\{(R,H),(R,T),(B,H),(B,T),(G,H),(G,T)\}$$

There are \(3 \times 2 = 6\) outcomes.

7. Worked Example 4: Check if a list is complete

Question: Jamal rolls a number cube and flips a coin. He writes this sample space:

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

Is his sample space complete?

No, it is not complete.

He listed all outcomes with heads, but for tails he only listed \((1,T), (2,T), (3,T)\).

He is missing:

• \((4,T)\)
• \((5,T)\)
• \((6,T)\)

The complete sample space should have \(6 \times 2 = 12\) outcomes.

8. How to know if your sample space is exhaustive

Ask yourself these questions:

• Did I include every outcome from the first event?
• Did I match each one with every outcome from the second event?
• Did I keep the order correct?
• Did I count the total number of outcomes?

A quick way to check is to multiply:

$$\text{number of outcomes in first event} \times \text{number of outcomes in second event}$$

If the first event has \(a\) outcomes and the second event has \(b\) outcomes, then the compound event has:

$$a \times b$$

total outcomes.

For example:

• coin and number cube: \(2 \times 6 = 12\)
• number cube twice: \(6 \times 6 = 36\)
• 3-color spinner and coin: \(3 \times 2 = 6\)

9. Common mistakes to avoid

• Forgetting outcomes — make your list organized so you do not skip any.
• Repeating outcomes — check that each pair appears only once.
• Ignoring order — in sequential events, the order matters.
• Not checking the total — multiply to see how many outcomes there should be.

10. Summary

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

A compound event has two or more actions, like flipping a coin and rolling a number cube.

You can find sample spaces for compound events by using an organized list or a table.

To make sure your sample space is complete, pair each outcome from the first event with every outcome from the second event and check the total by multiplying.

Tree Diagrams for Probability

Probability helps us describe how likely something is to happen. Some chance events happen in more than one step. For example, you might flip a coin and then roll a die. When an event has stages, it can be hard to list all the possible outcomes in an organized way.

A tree diagram is a picture that helps us show all the possible outcomes of a multi-step chance event. It helps us count outcomes carefully so we do not miss any and do not count any twice.

In this lesson, you will learn how to read a tree diagram, draw your own tree diagram, and use it to find probabilities.

1. What is a tree diagram?

A tree diagram starts with one point. From that point, we draw branches to show the possible results of the first step. Then, from each of those branches, we draw more branches to show the possible results of the next step.

Each path from the start to the end shows one complete outcome.

For example, if you flip a coin, the outcomes are:

• Heads
• Tails

If you flip a coin twice, the first flip has 2 choices, and then the second flip has 2 choices again. A tree diagram helps us list all the outcomes:

• HH
• HT
• TH
• TT

There are 4 possible outcomes in total.

2. How to make a tree diagram

To draw a tree diagram, follow these steps:

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

Important: Each stage must show all possible results. Be careful to branch from every result in the earlier stage.

3. Using tree diagrams to find probability

Once the tree diagram shows all possible outcomes, we can find probability by using this rule:

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

A favorable outcome is an outcome that matches the event we want.

For example, if there are 6 possible outcomes and 2 of them are what we want, then:

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

So the probability is \(\frac{1}{3}\).

4. Worked Example 1: Flipping a coin twice

Question: A coin is flipped two times. What is the probability of getting exactly one head?

Step 1: Draw or imagine the tree diagram.

• First flip: H or T
• Second flip after H: H or T
• Second flip after T: H or T

The complete outcomes are:

• HH
• HT
• TH
• TT

Step 2: Find the favorable outcomes.

Exactly one head means one H and one T. The favorable outcomes are:

• HT
• TH

So there are 2 favorable outcomes out of 4 total outcomes.

Step 3: Write the probability.

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

Answer: The probability of getting exactly one head is \(\frac{1}{2}\).

5. Worked Example 2: Rolling a die and flipping a coin

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

Step 1: Show the first stage.

The die can land on:

• 1, 2, 3, 4, 5, 6

Step 2: Show the second stage.

After each die result, the coin can land on:

• H
• T

Step 3: Count all outcomes.

Each of the 6 die results connects to 2 coin results, so there are:

$$6 \times 2 = 12$$

possible outcomes.

We can list some of them:

• 1H, 1T
• 2H, 2T
• 3H, 3T
• 4H, 4T
• 5H, 5T
• 6H, 6T

Step 4: Find favorable outcomes.

Even numbers are 2, 4, and 6. We want tails, so the favorable outcomes are:

• 2T
• 4T
• 6T

There are 3 favorable outcomes.

Step 5: Write the probability.

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

Answer: The probability is \(\frac{1}{4}\).

6. Worked Example 3: Choosing from colors

Question: A spinner has 3 equal sections: red, blue, and green. It is spun twice. What is the probability of getting red both times?

Step 1: First spin outcomes.

• R
• B
• G

Step 2: Second spin outcomes from each branch.

• R, B, G

Step 3: List all complete outcomes.

• RR
• RB
• RG
• BR
• BB
• BG
• GR
• GB
• GG

There are 9 total outcomes.

Step 4: Find the favorable outcomes.

Getting red both times means only:

• RR

So there is 1 favorable outcome.

Step 5: Write the probability.

$$\frac{1}{9}$$

Answer: The probability of getting red both times is \(\frac{1}{9}\).

7. Worked Example 4: A two-stage choice with words

Question: A lunch special lets you choose 1 sandwich and 1 drink.

• Sandwiches: cheese or tuna
• Drinks: juice, water, or milk

If each choice is equally likely, what is the probability of choosing tuna and milk?

Step 1: First stage

• Cheese
• Tuna

Step 2: Second stage from each sandwich

• Juice
• Water
• Milk

Step 3: List all outcomes.

• Cheese + Juice
• Cheese + Water
• Cheese + Milk
• Tuna + Juice
• Tuna + Water
• Tuna + Milk

There are 6 possible outcomes.

Step 4: Find the favorable outcome.

We want:

• Tuna + Milk

There is 1 favorable outcome.

Step 5: Write the probability.

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

Answer: The probability is \(\frac{1}{6}\).

8. A useful pattern

In many tree diagrams, the total number of outcomes can be found by multiplying the number of choices at each stage.

For example:

• 2 coin results and 2 coin results gives \(2 \times 2 = 4\) outcomes
• 6 die results and 2 coin results gives \(6 \times 2 = 12\) outcomes
• 3 spinner results and 3 spinner results gives \(3 \times 3 = 9\) outcomes

This works because each choice in one stage connects to all the choices in the next stage.

9. Common mistakes to avoid

• Forgetting branches: Make sure every first-stage result has all the second-stage branches.
• Missing outcomes: Check each full path from start to finish.
• Counting the wrong favorable outcomes: Read the question carefully. For example, “exactly one head” is not the same as “at least one head.”
• Not simplifying fractions: If possible, write the probability in simplest form.

10. How to check your work

After finishing a tree diagram, ask yourself:

• Did I show all the results for the first stage?
• Did I branch correctly for the next stage?
• Did I count all complete outcomes?
• Did I count only the outcomes the question asks for?
• Did I simplify my fraction?

11. Quick practice thinking

If you toss a coin and then choose a card labeled A or B, the outcomes are:

• HA
• HB
• TA
• TB

If you want the probability of getting tails and B, there is 1 favorable outcome out of 4 total outcomes, so the probability is:

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

This is exactly how tree diagrams help: they turn a complicated event into an organized list.

Summary

A tree diagram is a tool for showing all possible outcomes in a chance event with more than one step. Each branch shows a possible result, and each full path shows one complete outcome.

To find a probability with a tree diagram, count the total number of outcomes and then count the favorable outcomes. Then use:

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

When you draw carefully and count carefully, tree diagrams make probability much easier to understand.