Tree Diagram

Probability and Tree Diagram

Probability is a Mathematical term that expresses the likelihood of the occurrence of an event. Probability is used intensively in a lot of industries and rigorously in Mathematical calculation.

A tree diagram in probability is a visual representation of a hierarchy in a tree-like structure. A tree diagram consists of elements like the root node or parent node. There are some nodes linked with branches, which represent the relationship between nodes.  Further, there are nodes linked with branches. They are called leaf nodes or end nodes.

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Root Node

The root node has no superior parent. It itself is the parent node. 


Nodes are the family members of the root node linked with the help of branches and express the relationship among the members. 


The nodes which are linked together to represent relationships and connections among them with the help of lines joining the nodes are called branches.

Leaf Nodes

Leaf nodes are also called child nodes or end nodes because they are the last members of a tree diagram. They have no further child node or children.

Where are Tree Diagram Examples are Used: 

  • To show family relations.

  • In mathematics.

  • In the computer science field.

  • In taxonomy.

  • In business organizations.

  • In science of classification.

Generally, a tree diagram is used in the classification of things or to represent a sequence of events. A tree diagram starts with a parent node and branches into two or more nodes. Each node will further branch into more nodes and so on. This begins to resemble the structure of a tree. In mathematics, tree diagrams are used in statistics and probability.

Tree Diagram in Probability

In probability theory, a probability tree diagram shows all the possible outcomes. The first event is represented by a dot. Branches emerging from this dot represent all the possible outcomes. The probability of each outcome is written on its branch until a conclusion is reached.

The probability of any event tells us how likely to happen something is.   In other words, it tells us about the possibility of an event taking place. If a coin is tossed, what is the sure outcome of this event?

There are two favorable outcomes, one is head and the other is tail. So the best to know how likely they are to occur is by using the probability theory.

The probability value is a numerical value and it always lies between 0 and 1. The probability of an impossible event is zero and the probability of the sure event is 1.

The formula of probability is given below

Probability of an event, P(E) = \[\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\].

Regarding a tree diagram, kindly take the following into consideration:

  1. The tree diagram should include all the possibilities of an outcome

  2. The sum of all the possibilities should be 1.

  3. The number of branches shown in the tree diagram actually represents the number of different possibilities

  4. Probabilities have to be represented by writing them on the branches of the tree. 

Calculating Overall Probability and Probability Tree Diagram

Suppose we toss a coin two times, what will be the sample space?

If we toss a coin twice, the favorable outcomes will be ahead or a tail. The possible outcomes may be

Heads in both the attempts= (H,H)=HH

Head in the first attempt and Tail in the second attempt=(H,T)=HT

Tail in the first attempt and head in the second attempt=(T,H)=TH

Tail in both the attempts=(T,T)=TT

So the sample space of the event will be {HH,HT,TH,TT}.

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Steps to Calculate Overall Probability

Write the probability value on the branches. Multiply the probability value along the branches and add the probability value obtained after the multiplication.

When we add all the probability values, the result should be equal to 1.

Tree Diagram Example:

Question 1: A bag contains 3 black and 2 white balls. George picks a ball at random from the bag and puts it back in the bag. He mixes the balls in the bag and picks another ball at random from the same bag.

1- Construct a probability tree diagram.

2- Using the tree diagram, calculate the probability of picking two black balls.


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Check that the probabilities in the last column add up to 1.

2) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. Since these are independent events we can multiply the probability of each branch.

So, the probability of getting two black balls= ⅗ × ⅗ = 9/25

Questions for Practice:

  1. From an ordinary deck of 52 playing cards, a card is picked. Then, without replacement, another card is picked. Draw a probability tree for the above-mentioned situation and using it, try to calculate the probability of picking two black cards.

  2. Joy has been assigned some work in Science and English. His probability of completing Science work on any given day is 1/3. The probability that he will complete his English work is ¼. Both of these are independent events. 

By drawing a tree diagram, calculate the following: 

  1. The probability of Joy completing his work in both the subjects

  2. The probability that Joy will complete the homework of only one subject on a given day. 

FAQ (Frequently Asked Questions)

1. What is the Sample Space?


A sample space is the set of all possible outcomes in an experiment such as choosing a ball, drawing a card, etc). It is usually denoted by the letter S. Sample space can be written using the set notation { }.

The sum of all the probabilities of the distinct outcomes within a sample space is 1.

2. What are the Applications of Probability in the Real World?

Solution: A basic knowledge of probability simplifies the understanding of several other phenomena around us. Probability is used in weather forecasting, sports strategies, considering insurance schemes and plans, etc.