Tree Diagram

A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They can help you record all the possibilities in a clear, uncomplicated manner. Each branch in a tree diagram represents an outcome. Consider the following example:

Example

If we tossed a fair coin twice, the first outcome would not affect the second outcome (the two events are independent). The outcome of the first toss can either be a head or a tail. Similarly, the outcome of the second toss can be a head or a tail, regardless of the outcome of the first toss, since the coin has no memory! We can represent the probabilities in a tree diagram as shown below.

Please note the following:

• the tree diagram must include all the possible outcomes;
• the sum of the probabilities must add up to 1;
• the number of branches is the number of different possibilities; and
• probabilities are represented by the numbers on the branches.

To calculate probabilities, we go along the tree branches from left to right to get to the end and then multiply any probabilities that we have passed together. Therefore, the probability of getting two heads, i.e., HH, is:

$$ \begin{align*}
\text P(\text{HH})& =\cfrac {1}{2} * \cfrac {1}{2} \\
&= \cfrac {1}{4} \\
\end{align*} $$

If we sum up the probabilities of all possibilities, we get 1. (The candidate can prove this.)

Question

A card is picked from an ordinary pack of 52 playing cards, without replacement, and then another one is picked. Draw a probability tree and use it to calculate the probability of picking two red cards.

A. 25/102

B. 13/51

C. 26/51

Solution

The correct answer is A.

Reading 8 LOS 8j

Explain the use of a tree diagram to represent an investment problem.