Probability Tree Diagram
A probability tree diagram is a diagram that is used to give a visual representation of the probabilities as well as the outcomes of an event. A probability tree diagram consists of two parts - nodes and branches. A node is used to represent an event. A branch is used to denote the connection between an event and its outcome.
A probability tree diagram can be used to depict conditional probabilities as well as independent events. In this article, we will learn more about a probability tree diagram, how to draw a probability tree, and see some examples.
|1.||What is a Probability Tree Diagram?|
|2.||Probability Tree Diagram Example|
|3.||Conditional Probability Tree Diagram|
|4.||How to Draw a Probability Tree?|
|5.||FAQs on Probability Tree Diagram|
What is a Probability Tree Diagram?
A probability tree diagram is used to represent the probability of occurrence of events without using complicated formulas. It displays all the possible outcomes of an event. The purpose of a probability tree is that it shows all the possible outcomes of an event and calculates the probability of these outcomes. A probability tree diagram can either represent a series of independent events or it can be used to denote conditional probabilities.
Parts of a Probability Tree Diagram
There are two main parts of a probability tree. These are the nodes and the branches. The nodes can further be classified into a parent node and a sibling node. The parent node represents a certain event and has a probability of 1. The sibling nodes denote other additional possible events or outcomes. The branches denote the probability of occurrence of these events. Suppose a fair coin is tossed once, then the probability tree can be constructed as follows:
This is a simple probability tree and has two branches only. Here, the first node represents the parent event of a coin being tossed. Head and tail are the two possible outcomes forming the sibling nodes. 0.5 is written on the branch and represents the probability of occurrence of each sibling event.
Probability Tree Diagram Example
Suppose a probability tree diagram needs to be constructed for flipping a fair coin twice. This is an example of an independent event as the outcome of each coin toss is independent of the previous flip. First, the probability tree diagram of a coin being flipped once is drawn as given in the previous section. The next step is to extend it to two coin tosses as follows:
The second set of probabilities represents the second coin toss. Thus, in total there are 4 sets of possible outcomes.
To calculate the probabilities of a series of events, multiply the probabilities along the branches of the probability tree diagram. The total probability can be computed by adding these probabilities and its value will always be equal to 1.
Some useful inferences can be made from the probability tree diagram as follows:
- The probability of getting the outcome (Head, Head) = 0.5 × 0.5 = 0.25. Similarly, the probability of the other outcomes can be calculated.
- 0.25 + 0.25 + 0.25 + 0.25 = 1. This implies that on adding the probabilities of each outcome, the total is equal to 1.
- By looking at the probability tree, the probability of getting exactly one head can be calculated as 0.25 + 0.25 = 0.5.
Conditional Probability Tree Diagram
A conditional probability tree diagram is very useful in depicting the outcome of dependent events. A dependent event is one whose outcome is affected by an event that has previously occurred. Suppose a student has to appear for two tests. The probability that he will pass the first test is 0.7. If he passes the first test, the probability that he passes the second test is 0.8. Furthermore, if he fails the first test, the probability that he passes the second test is 0.6. To determine the conditional probability that he fails in both tests, the probability tree diagram can be constructed using the following steps.
- Step 1: Construct the probability tree diagram for the first test. As the probability of passing the first test is 0.7 thus, the probability of failure is 1 - 0.7 = 0.3.
- Step 2: Construct branches for the second test given that the student passes the first test. It must be noted that the sum of probabilities of outcomes, 0.8 + 0.2, is 1.
- Step 3: Now, draw branches for the second test given that the student fails the first test.
- Step 4: Multiply the probabilities along each branch of the probability tree to get the conditional probability results.
From the probability tree diagram, it can be concluded that the conditional probability for a student who fails the second test given that he fails the first is 0.12.
How to Draw a Probability Tree?
To draw a probability tree diagram it is necessary to identify all the possible outcomes and the probabilities associated with them. The steps to construct a probability tree are as follows:
- Step 1: Identify whether the events are dependent or independent.
- Step 2: Draw branches to represent the first set of outcomes.
- Step 3: Write the probabilities associated with each outcome on the branch.
- Step 4: Draw the next set of branches taking into account whether the subsequent events are dependent or independent. Also, write the associated probabilities.
- Step 5: Repeat step 5 for as many branches as required.
- Step 6: To calculate the overall probability of the probability tree, the probabilities along the branches have to be multiplied. The sum of these probabilities should always be equal to 1.
Important Notes on Probability Tree Diagram
- A probability tree diagram is used to visually represent all the possible outcomes of an event along with the associated probabilities.
- Nodes and branches make up the parts of a probability tree diagram. The node represents an event and their occurence probabilities are written on the branches.
- Multiply the probabilities along the branches to calculate the probability of outcomes.
- Add the probabilities to get the probability of multiple outcomes.
- The sum of the probabilities of all possible outcomes of a probability tree diagram is always 1.
Examples on Probability Tree Diagram
Example 1: Suppose a bag contains a total of 5 balls out of which 2 are blue and 3 are orange. Using a probability tree diagram, find the conditional probability of drawing a blue ball given the first ball that was drawn was orange. The balls are drawn without replacement.
Solution: The probability tree can be constructed as follows:
The chance of picking out an orange ball on the first draw is 3 / 5.
Now the total number of balls remaining in the bag after the first draw is 4.
The probability of picking a blue ball on the second draw is 2 / 4
From the probability tree diagram, the probability of picking a blue ball given that an orange ball has been drawn is
= (3 / 5) × (2 / 4) = 3 / 10
Answer: The conditional probability is 3 / 10
Example 2: The letters of the word "CHICKEN" are written on 7 cards. Kelly chooses a card and replaces it and chooses another one. Using the probability tree diagram find the probability that only one of the chosen cards will have the letter C on it.
Solution: Let C represent the event of getting the letter C. C' denotes the event of not getting the letter C. The probability tree diagram can be drawn as follows:
Thus, P(C, C') + P(C',C) = (2 / 7)(5 / 7) + (5 / 7)(2 / 7) = 20 / 49
Answer: The probability of getting a C on only one of the cards is 20 / 49.
Example 3: A fair coin is flipped thrice. Using the probability tree diagram find the probability of getting at least two tails.
Solution: The probability tree diagram for a fair coin being flipped three times is given as follows:
From the probability tree we get,
P (at least two tails) = P(H, T, T) + P(T, H, T) + P(T, T, H) + P(T, T, T) = (1/ 8) + (1/ 8) + (1/ 8) + (1/ 8)
= 4 / 8 = 1/2
Answer: The probability of getting at least two tails on flipping a coin thrice is 1 / 2.
FAQs on Probability Tree Diagram
What is a Probability Tree Diagram in Probability Theory?
A probability tree diagram is used to give a visual representation of the probability of occurrences of all possible outcomes of an event. It can be used to demonstrate dependent and independent events.
What are the Parts of a Probability Tree Diagram?
There are two parts to a probability tree diagram. These are given as follows:
- Node - it is used to represent an event.
- Branch - it is used to connect two events. The probability of occurrence of an event is written on a branch.
How to Calculate the Probability Using a Probability Tree Diagram?
By multiplying the probabilities of connecting branches the probability of a series of events can be calculated. The sum of all these probabilities must be equal to 1.
What is a Conditonal Probability Tree Diagram?
A conditional probability tree diagram is used to represent conditional probabilities. In other words, this probability tree can be used to find the probabilities of outcomes of dependent events.
How To Draw a Probability Tree?
The steps to draw a probability tree are as follows:
- Draw branches of the first set of outcomes and write the individual probabilities along the branches.
- Repeat this process for the remaining outcomes.
- Multiply the values of probabilities of connecting branches to get the likelihood of occurrence of each outcome.
Can the Sum of Probabilities of Outcomes of a Probability Tree Diagram be Negative?
The value of a probability will always lie between 0 and 1. Thus, the probability of outcomes of a probability tree diagram will always be positive. Furthermore, the sum of all probabilities will always be equal to 1.
How to Calculate the Probability of Multiple Events Using a Probability Tree Diagram?
The first step is to find the probability of each outcome by multiplying the probability values of connecting branches in the probability tree diagram. Next, add the probabilities associated with the required events to determine the probability of multiple events.