How to use a tree diagram

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.


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

Watch this video on YouTube.

Can’t see the video? Click here.

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!


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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


Tree diagram

Tree diagram diagram is a tool that allows examine the subject systematically (problem) in the form of constituent elements (reasons) and show logical connections between these elements. treelike the diagram is built in the form of a multi-stage tree structure, component parts which are the various elements of the solution Problems. The principle of constructing a tree diagram is illustrated in the figure.

Tree diagram is used to identify and showing the connection between the subject (problem) review and its components (elements, causes), for example, in when:

• unclear formulated wishes of the consumer in relation to products are converted first in the established and expected needs and then to technical conditions for this product;

• necessary explore all possible parts, concerning the subject under consideration;

• short-term goals must be achieved before results of all work.

Matrix diagram.

Matrix diagram - a tool to identify the importance various connections. matrix chart used for such an organization and presentation of a large number data (elements) to graphically illustrate logical connections between different elements while simultaneously showing the importance of these connections. Target matrix chart - spreadsheet representation of logical connections and the relative importance of these links between a lot of verbal descriptions related to the following:

•tasks (problems) of quality;

• reasons quality problems;

• requirements, established and expected consumer needs;

• characteristics and product features;

• characteristics and process functions;

• characteristics and functions of production operations and equipment.

Example matrix diagram is given in table. 4.2.

20 . Implementation Process Diagram programs

Diagram Program Implementation Process (PDPC) - tool used for graphic representation sequence of actions and decisions, necessary to achieve the set goals.

Graphic PDPC representation:

Usually PDPC is used to evaluate the timing and the feasibility of performing work in according to the Gantt chart or network diagram to correct them. In addition, the implementation process diagram programs are easy to use improvement research process, through the accumulation of detailed data on its actual course, and as well as identifying potential problems implementation of the process is still at the stage its design.

21 By the mid-1980s, most widespread "waterfall" or "cascading" creation process software. His main the characteristic is the partition of the entire development into stages, and the transition from one stage to the next only after it is completely completed work on the current one. Each stage ends with the release of the complete a set of documentation sufficient in order for development to continue another development team.

Application "cascade" process is effective for systems for which at the very beginning development can be sufficiently accurate and fully formulate all requirements, to provide developers the freedom to realize them as best as possible from a technical point of view. In this category includes complex settlement real time systems and others similar tasks. However, in the process using this approach, it turned out a number of its shortcomings. In process software, there was a constant need for return to previous steps and clarification or reconsideration of previous decisions. As a result, the real process of creating system took the following form:

The process has a number of essential shortcomings, the main of which is, perhaps, that the requirements for created system are "frozen" in the form of a technical task for all time her creation. Thus, users can only comment after work on the system will be fully completed. When inaccurate statement of requirements or their changes over a long period system creation, users receive system that does not satisfy them needs.

22 In real life, it turns out that on requirements formulation stage cannot accurately define all requirements to the software product. To overcome this problem in the second half of the 80s years, a "spiral" process was proposed program development with an emphasis on stages of analysis and design. Development systems according to this methodology occurs iterations, and the passage of each turn helix user gets regular system version. Upon receipt by the customer each version clarifies the goals and characteristics of the project, determined its quality and planned work next turn of the spiral.

Spiral software development model, in one way or another versions used in many specific applied techniques, built on the next template. First of all, in during communication with the customer is determined set of the most important and critical possibilities of the future system. Further determined by joint efforts the desired time frame for the implementation of this basic functionality. Formed plan, work begins, and is tracked their implementation.

AT The spiral model is based on two parcels. Numerous studies confirmed that both the customer and the contractor usually overly optimistic to deadlines and budget, even when using good methodologies for estimating the scope of work. Therefore, the results of such estimates it is proposed to increase (deteriorate) quite seriously - by about 50%. In addition, the customer is usually weak represents the architecture of the future system, so it should be designed invest in open technologies and maximum flexibility for expansion and increasing functionality.

23 Formal systems development model based on the development of a formal mathematical software specification system and transformations of this specification through special mathematical methods into executable programs. Examination compliance with specifications and system components is also performed mathematical methods.

Ticket 24

Development software based previously created components

AT most software projects reuse applies some software modules. it usually happens where developers project know about previously created software products that include there are components, approximately meeting the requirements of the developed components. These components are modified according to new requirements, and only after that is included in the composition new system. In the evolutionary model developments to speed up the process software creation reuse previously created components are applied often enough.

This approach is based on the presence of a large base existing software components, which can be integrated into the new system. Often these components are freely traded on the market software products that can use to perform certain special features such as text formatting, numeric calculations, etc.

AT this approach, the initial stage of specification requirements and certification stage are the same, as in other models of the creation process ON. And the steps in between have the following meaning.

a) component analysis. Having a specification requirements, at this stage search for components that could satisfy the formulated requirements. Usually impossible to accurately match functions implemented ready-made components, and functions, defined by the requirements specification.

b) Requirements modification. At this stage requirements are analyzed taking into account information about the components obtained at the previous stage. Requirements modified in such a way that make the most of the opportunities selected components. If the change requirements is not possible, repeatedly component analysis is performed for to find some alternative solution.

in) System design. At this stage the structure of the system is designed or the existing structure is modified reusable system. The design must take into account selected software components and build a structure according to their functionality. If a some ready-made software components unavailable, "new" software is being designed.

G) Development and assembly of the system. This is the stage direct creation of the system. AT within the framework of the considered approach, the assembly system is more of a development system than as a separate step. See fig. 1.5.

posted at http://www.

Picture 1.5 - Software development with reuse previously created components

Main advantages of the described process model software development with reuse previously created components are that the number of directly developed components and reduce the cost the system being created.

Together so when using this approach compromises are inevitable in determining requirements; this may lead to that the complete system will not meet all customer requirements. In addition, when upgrading system (i.e. when creating its new version) there is no possibility to influence to the appearance of new versions of components, used in the system, which is significantly complicates the modernization process.

Models incremental development and spiral software development has its advantages and disadvantages. Mainly applied when creating large information systems and usually use different approaches to developing different parts of the system, i.e. in general to the development systems apply hybrid models.

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