How to draw probability tree diagrams

Probability Tree Diagrams: Examples, How to Draw

Probability > How to Use a Probability Tree

Probability trees are useful for calculating combined probabilities for sequences of events. It helps you to map out the probabilities of many possibilities graphically, without the use of complicated probability formulas.

Watch the video for an example.

How to draw a probability tree

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Why Use a probability tree?
Sometimes you don’t know whether to multiply or add probabilities. A probability tree makes it easier to figure out when to add and when to multiply. Plus, seeing a graph of your problem, as opposed to a bunch of equations and numbers on a sheet of paper, can help you see the problem more clearly.

Parts of a Probability Tree Diagram

A probability tree has two main parts: the branches and the ends(sometimes called leaves). The probability of each branch is generally written on the branches, while the outcome is written on the ends of the branches.

Multiplication and Addition
Probability Trees make the question of whether to multiply or add probabilities simple: multiply along the branches and add probabilities down the columns. In the following example (from Yale University), you can see how adding the far right column adds up to 1, which is what we would expect the sum total of all probabilities to be:
.9860 + 0.0040 + 0.0001 + 0.0099 = 1

Real Life Uses

Probability trees aren’t just a theoretical tool used the in the classroom—they are used by scientists and statisticians in many branches of science, research and government. For example, the following tree was used by the Federal government as part of an early warning program to assess the risk of more eruptions on Mount Pinatubo, an active volcano in the Philippines.
Image: USGS.

How to Use a Probability Tree or Decision Tree

Sometimes, you’ll be faced with a probability question that just doesn’t have a simple solution. Drawing a probability tree (or tree diagram) is a way for you to visually see all of the possible choices, and to avoid making mathematical errors. This how to will show you the step-by-step process of using a decision tree.
How to Use a Probability Tree: Steps
Example question: An airplane manufacturer has three factories A B and C which produce 50%, 25%, and 25%, respectively, of a particular airplane. Seventy percent of the airplanes produced in factory A are passenger airplanes, 25% of those produced in factory B are passenger airplanes, and 25% of the airplanes produced in factory C are passenger airplanes. If an airplane produced by the manufacturer is selected at random, calculate the probability the airplane will be a passenger plane.

Step 1:Draw lines to represent the first set of options in the question (in our case, 3 factories). Label them: Our question lists A B and C so that’s what we’ll use here.

Step 2: Convert the percentages to decimals, and place those on the appropriate branch in the diagram. For our example, 50% = 0.5, and 25% = 0.25.

Step 3: Draw the next set of branches. In our case, we were told that 70% of factory A’s output was passenger. Converting to decimals, we have 0.7 P (“P” is just my own shorthand here for “Passenger”) and 0.3 NP (“NP” = “Not Passenger”).

Step 4:Repeat step 3 for as many branches as you are given.

Step 5: Multiply the probabilities of the first branch that produces the desired result together. In our case, we want to know about the production of passenger planes, so we choose the first branch that leads to P.

Step 6: Multiply the remaining branches that give the desired result. In our example there are two more branches that can lead to P.

Step 6: Add up all of the probabilities you calculated in steps 5 and 6. In our example, we had:

. 35 + .0625 + .0625 = .475

That’s it!

Example 2

Example Question: If you toss a coin three times, what is the probability of getting 3 heads?

The first step is to figure out your probability of getting a heads by tossing the coin once. The probability is 0.5 (you have a 50% probability of tossing a heads and 50% probability of tossing a tails). Those probabilities are represented at the ends of each branch.

Next, add two more branches to each branch to represent the second coin toss. The probability of getting two heads is shown by the red arrow. To get the probability, multiply the branches:
0.5 * 0.5 = 0.25 (25%).
This makes sense because your possible results for one head and one tails is HH, HT, TT, or TH (each combination has a 25% probability).

Finally, add a third row (because we were trying to find the probability of throwing 3 heads). Multiplying across the branches for HHH we get:
0.5 * 0.5 * 0.5 = 0.125, or 12.5%.

In most cases, you will multiply across the branches to get probabilities. However, you may also want to add vertically to get probabilities. For example, if we wanted to find out our probability of getting HHH OR TTT, we would first calculated the probabilities for each (0.125) and then we would add both those probabilities: 0.125 + 0.125 = 0.250.

Tip: You can check you drew the tree correctly by adding vertically: all the probabilities vertically should add up to 1.

Next: Tree Diagram Real Life Example


Punongbayan, R. et al. USGS Repository: Eruption Hazard Assessments and Warnings.

Stephanie Glen. "Probability Tree Diagrams: Examples, How to Draw" From Elementary Statistics for the rest of us! https://www.


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Probability Tree Diagrams Explained! — Mashup Math

This quick introduction will teach you how to calculate probabilities using tree diagrams.

Figuring out probabilities in math can be confusing, especially since there are many rules and procedures involved. Luckily, there is a visual tool called a probability tree diagram that you can use to organize your thinking and make calculating probabilities much easier.

At first glance, a probability tree diagram may seem complicated, but this page will teach you how to read a tree diagram and how to use them to calculate probabilities in a simple way. Follow along step-by-step and you will soon become a master of reading and creating probability tree diagrams.

What is a Probability Tree Diagram?

Example 01: Probability of Tossing a Coin Once

Let’s start with a common probability event: flipping a coin that has heads on one side and tails on the other:

This simple probability tree diagram has two branches: one for each possible outcome heads or tails. Notice that the outcome is located at the end-point of a branch (this is where a tree diagram ends).

Also, notice that the probability of each outcome occurring is written as a decimal or a fraction on each branch. In this case, the probability for either outcome (flipping a coin and getting heads or tails) is fifty-fifty, which is 0.5 or 1/2.

Example 02: Probability of Tossing a Coin Twice

Now, let’s look at a probability tree diagram for flipping a coin twice!

Notice that this tree diagram is portraying two consecutive events (the first flip and the second flip), so there is a second set of branches.

Using the tree diagram, you can see that there are four possible outcomes when flipping a coin twice: Heads/Heads, Heads/Tails, Tails/Heads, Tails/Tails.

And since there are four possible outcomes, there is a 0.25 (or ¼) probability of each outcome occurring. So, for example, there is a 0.25 probability of getting heads twice in a row.

How to Find Probability

The rule for finding the probability of a particular event in a probability tree diagram occurring is to multiply the probabilities of the corresponding branches.

For example, to prove that there is 0.25 probability of getting two heads in a row, you would multiply 0.5 x 0.5 (since the probability of getting a heads on the first flip is 0.5 and the probability of getting heads on the second flip is also 0.5).

0.5 x 0.5 = 0.25

Repeat this process on the other three outcomes as follows, and then add all of the outcome probabilities together as follows:

Note that the sum of the probabilities of all of the outcomes should always equal one.

From this point, you can use your probability tree diagram to draw several conclusions such as:

·       The probability of getting heads first and tails second is 0.5x0.5 = 0.25

·       The probability of getting at least one tails from two consecutive flips is 0.25 + 0.25 + 0.25 = 0.75

·       The probability of getting both a heads and a tails is 0.25 + 0.25 = 0.5

Independent Events and Dependent Events

What is an independent event?

Notice that, in the coin toss tree diagram example, the outcome of each coin flip is independent of the outcome of the previous toss. That means that the outcome of the first toss had no effect on the probability of the outcome of the second toss. This situation is known as an independent event.

 What is a dependent event?

Unlike an independent event, a dependent event is an outcome that depends on the event that happened before it. These kinds of situations are a bit trickier when it comes to calculating probability, but you can still use a probability tree diagram to help you.

Let’s take a look at an example of how you can use a tree diagram to calculate probabilities when dependent events are involved.

How to Make a Tree Diagram

Example 03:

Greg is a baseball pitcher who throws two kinds of pitches, a fastball, and a knuckleball. The probability of throwing a strike is different for each pitch:

·       The probability of throwing a fastball for a strike is 0.6

·       The probability of throwing a knuckleball for a strike 0.2

Greg throws fastballs more frequently than he throws knuckleballs. On average, for every 10 pitches he throws, 7 of them are fastballs (0.7 probability) and 3 of them are knuckleballs (0. 3 probability).

So, what is the probability that the pitcher will throw a strike on any given pitch?

 To find the probability that Greg will throw a strike, start by drawing a tree diagram that shows the probability that he will throw a fastball or a knuckleball

The probability of Greg throwing a fastball is 0.7 and the probability of him throwing a knuckleball is 0.3. Notice that the sum of the probabilities of the outcomes is 1 because 0.7 + 0.3 is 1.00.

 Next, add branches for each pitch to show the probability for each pitch being a strike, starting with the fastball:

Remember that the probability of Greg throwing a fastball for a strike is 0.6, so the probability of him not throwing it for a strike is 0.4 (since 0.6 + 0.4 = 1.00)

Repeat this process for the knuckleball:

Remember that the probability of Greg throwing a knuckleball for a strike is 0. 2, so the probability of him not throwing it for a strike is 0.8 (since 0.2 + 0.8 = 1.00)

Now that the probability tree diagram has been completed, you can perform your outcome calculations. Remember that the sum of the probability outcomes has to equal one:

Since you are trying to figure out the probability that Greg will throw a strike on any given pitch, you have to focus on the outcomes that result in him throwing a strike: fastball for a strike or knuckleball for a strike:

The last step is to add the strike outcome probabilities together:

0.42 + 0.06 = 0.48

 The probability of Greg throwing a strike is 0.48 or 48%.

Probability Tree Diagrams: Key Takeaways

·      A probability tree diagram is a handy visual tool that you can use to calculate probabilities for both dependent and independent events.

·      To calculate probability outcomes, multiply the probability values of the connected branches.

·      To calculate the probability of multiple outcomes, add the probabilities together.

·      The probability of all possible outcomes should always equal one. If you get any other value, go back and check for mistakes.


Check out the animated video lessons and keep

Check out the video lessons below to learn more about how to use tree diagrams and calculating probability in math:

Have thoughts? Share your input in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.


Create a tree diagram in Office

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A tree chart provides a hierarchical view of data and makes it easy to identify patterns, such as which items will perform best in a store. The branches of the tree are represented by a rectangle, and each branch is represented as a smaller rectangle. In a tree chart, categories are displayed by color and proximity, and can easily display large amounts of data, which would be difficult to do with other types of charts.

The tree chart is useful when you want to compare proportions in a hierarchy, but it doesn't show the hierarchical levels between the largest categories and each data point very well. The "sunburst" diagram is much more suitable for this.

Creating a tree diagram

  1. Select data.

  2. Go to tab Insert > insert hierarchical diagram > tree.

    Featured charts can also be used to create a tree chart, on the tab Insert > Featured charts > all charts.

Tip: On the tab Designer and Format you can customize the appearance of the chart. If you don't see these tabs, click anywhere in the tree view to activate them.

Changing how labels are displayed

Excel automatically uses a different color for each top-level (parent) category. But you can further highlight the differences between categories with the data label layout.

  1. Right-click one of the rectangles in the chart and select Data Series Format .

  2. Under Row Options > Signature Options , select the desired display option.

Creating a tree diagram

  1. Select data.

  2. On the ribbon, on the Insert tab, click the button (hierarchy icon) and select tree .

    Note: On the tab Designer and Format you can customize the appearance of the chart. If you don't see these tabs, click anywhere in the tree view to activate them.

Additional information

You can always ask the Excel Tech Community a question or ask for help in the Answers community.

See also

Create a Waterfall Chart

Creating a Pareto Chart

Create a histogram

Create a box and whisker chart

Create a sunburst chart in Office

Using Tree Diagrams - CyberPedia


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Although it is fairly easy to understand that the probability of getting heads on one flip of a fair coin is ½, it is somewhat more intuitive to determine the probability of four heads on four tosses of a fair coin. Although the coin example may seem artificial, it is well suited to explain the combination of probabilities over multiple attempts. Let's do the calculations. (Follow my reasoning, even if you are terribly afraid of mathematics. If you work through the examples, the calculations and mathematical reasoning will seem quite simple to you. No need to exclaim after looking at the next few numbers: “No, no way, I'll just skip it It is important to be able to think with numbers and about numbers.)

On the first roll, only one of two possible outcomes can occur; heads (O) or tails (P). What happens if a coin is tossed twice? There are four possible outcomes: heads both times (OO), heads the first time and tails the second time (OR), tails the first time and heads the second time (RO), and tails both times (RR). Since there are four possible outcomes and only one way of getting two heads, the probability of this event is 1/ 4 (again, we assume that the coin is “fair”, (312:) i. e. heads and tails are equally probable ). There is a general rule for calculating the probability of the joint occurrence of several events in any situation - the "and" rule. If you want to find the probability of the joint occurrence of the first and of the second event (heads on the first and on the second throw), you need to multiply the probabilities of these events separately. Applying the "and" rule, we find that the probability of getting two tails on a double coin toss is ½ x ½ = 1/ 4 . Intuitively, it seems that the probability of the joint occurrence of two events should be less than the probability of each of them separately; so it turns out.

A simple way to calculate this probability is obtained by representing all possible events with tree diagram. Tree diagrams were used in Chapter 4 when we tested the validity of "if...then..." statements. In this chapter, we will assign probabilistic values ​​to the branches of the tree to determine the probabilities of various combinations of outcomes. In later chapters, I will return to tree diagrams when I look at ways to find creative solutions to problems.

The first toss of a coin will land either heads or tails. For a "fair" coin, heads and tails have the same probability of 0.5. Let's picture it like this:

When you toss a coin a second time, either the first heads will be followed by a second heads or tails, or the first heads will be followed by a second heads or tails. The probabilities of getting heads and tails on the second toss are still 0.5. The outcomes of the second roll are shown on the diagram as additional branches of the tree.

As you can see from the diagram, there are four possible outcomes. You can use this tree to find the probabilities of other events. What is (313:) the probability of getting one head on two tosses of a coin? Since there are two ways to get one tail (OR or RO), the answer is 2 / 4 or ½. If you want to find the probability of two or more different outcomes, add up the probabilities of all outcomes. This is called the "or" rule. In another way, this problem can be formulated as follows: “What is the probability of getting or first heads, and then tails ( 1 / 4 ), or first tails, and then heads (1/4)?” The correct procedure for finding the answer is to add these values, resulting in ½. Intuitively, it seems that the probability of occurrence of one of several events should be greater than the probability of occurrence of each of them; so it turns out.

The rules "and" and "or" can be used only when the events of interest to us are independent. Two events are independent if the occurrence of one of them does not affect the occurrence of the other. In this example, the result of the first coin toss does not affect the result of the second toss. In addition, for the "or" rule to apply, the events must be incompatible, that is, they cannot occur at the same time. In this example, the outcomes are incompatible because we cannot get heads and tails on the same toss.

Tree diagram representation of events is useful in many situations. Let's expand our example. Suppose a man in a striped suit with a long, curled-up mustache and shifty little eyes stops you in the street and offers to play for money by tossing a coin. He always bets on the eagle. On the first toss, the coin lands heads up. The same thing happens on the second roll. On the third toss, heads come up again. When do you start to suspect that he has a "foul" coin? Most people have doubts on their third or fourth try. Calculate the probability of getting some heads on three and four fair coin tosses (the probability of getting heads is 0.5).

To calculate the probability of getting three heads in three attempts, you need to draw a tree with three rows of "nodes", with two "branches" coming from each node.

In this example, we are interested in the probability of getting three heads in a row, provided that the coin is fair. Look at the column labeled "outcome" and find the LLC outcome. Since this is the only outcome with three heads, multiply the probabilities along the 000 branch (circled in the diagram) and you get 0.5 x 0.5 x 0.5 = 0.125. A probability of 0.125 means that if the coin is "fair", then on average it will fall heads up three times in a row 12.5% ​​of the time. Since this probability is small, when three heads in a row fall out, most people begin to suspect that the coin is “with a secret”.

To calculate the probability of getting four heads in four attempts, add additional branches to the tree.

The probability of getting four heads is 0.5 x 0.5 x 0.5 x 0.5 = 0.0625, or 6.25%. As you already know, mathematically it is equal to 0.5 4 ; that is, multiplying a number by itself four times is the same as raising it to the fourth power. If you count on a calculator where there is an exponentiation operation, then you will get the same answer - 0.0625. Although such an outcome is possible and will someday happen, it is unlikely. In fact, he is so implausible and unusual that many would say that a person with shifty eyes is probably cheating. Undoubtedly, when the fifth eagle falls in a row, it will be reasonable to conclude0003

to say that you are dealing with a scammer. For most scientific purposes, an event is considered "unusual" if it is expected to occur with a probability of less than 5%. (In probability parlance, this is written as p < 0.05.)

Let's leave the artificial coin example and apply the same logic in a more useful context. I am sure that any student has ever come across multiple choice tests in which you need to choose the correct answers from the proposed options. In most of these tests, each question has five possible answers, of which only one is correct. Suppose the questions are so difficult that you can only randomly guess the correct answer. What is the probability of guessing correctly when answering the first question? If you have no idea which of the options is the correct answer, then you are equally likely to choose any of the five options, assuming that any of them could be correct. Since the sum of the probabilities of choosing all options should be equal to one, then the probability of choosing each of the options with the equiprobability of all options is 0.20. One of the options is correct and the rest are wrong, so the probability of choosing the correct option is 0.20. A tree diagram of this situation is shown below.

What is the probability of correctly guessing the answers to the first two questions of the test? We will have to add new branches to the tree, which will soon become very dense. To save space and simplify calculations, you can represent all incorrect options as a single branch, labeled "incorrect". The probability of making a mistake in answering one question is 0.8.

The probability of correctly guessing the answers to two questions is 0.2 x 0.2 = 0.04. That is, by chance it can happen only in 4% of attempts. Let's say we expand our example to three questions. I won't draw a tree, but you should already understand that the probability is 0. 2 x 0.2 x 0.2 = 0.008. This is such an unusual event that it can happen by chance in less than 1% of attempts. What do you think of a person who managed to answer all three questions correctly? Most people (and educators are people too) will conclude that the student did not choose answers at random, but actually knew something. Of course, it is possible that he was just lucky, but this is extremely unlikely. Thus, we come to the conclusion that the result obtained cannot be explained by luck alone.

I would like to point out one curious aspect of this reasoning. Consider the deplorable situation that Sarah found herself in. She answered 15 test questions, where the answer to each question had to be chosen from five options. Sarah answered all 15 questions incorrectly. Can you determine the probability that this happened by chance? I won't draw a tree diagram to illustrate this situation, but it's easy to see that the probability of answering one question wrong is 0.8; so the probability of answering all 15 questions incorrectly is 0. 8 15 . That number is 0.8 multiplied by itself 15 times, resulting in 0.0352. Since the probability of such an accident is 3.52%, maybe Sarah should tell the teacher that such an unusual result cannot be explained by chance? Sarah, of course, can make a similar argument, but would you believe her if you were a teacher? Suppose she claims to know the answers to all the questions. How else could she not choose the correct answer in 15 questions in a row? I don't know how many teachers would believe her claim that 15 wrong answers prove she has knowledge, although in principle such a line of reasoning is used to prove knowledge, since the probability of correctly guessing all the answers is about the same. (In this example, the probability of randomly answering all 15 questions correctly is 0.20 15 ; this number is well below 0.0001.) If I were Sarah's teacher, I would give her high marks for her creativity and understanding of statistical principles. It is possible that Sarah really knew something about this topic, but there was a systematic error in this “something”. I would also point out to her that she may not have prepared for the test, and in addition, she was also unlucky, and she made 15 wrong guesses. After all, sometimes very unusual things happen.

Before reading the next section, check that you understand how to use tree diagrams to calculate probabilities and account for all possible outcomes. I will return to such diagrams later in this chapter. Once you learn how to use them, you will be surprised how many situations they can be applied to.

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