Skill: Represent compound events using tree diagrams

A tree diagram shows the possible outcomes of a compound experiment in stages.

• Each branch represents one possible outcome.
• Write the probability on each branch.
• Multiply probabilities along one complete path.
• Add the probabilities of paths that make up the event you want.

Example

A bag contains 3 red counters and 2 blue counters. One counter is drawn, not replaced, and then a second counter is drawn. What is the probability of drawing one red and one blue?

Step 1: Draw the first branches

There are $5$ counters altogether:

$$P(R)=\frac{3}{5}, \qquad P(B)=\frac{2}{5}$$

The first branches are:

Start
├── Red     3/5
└── Blue    2/5

Step 2: Add the second-stage branches

Because the first counter is not replaced, the probabilities change.

• If the first counter is red, there are $2$ red and $2$ blue counters left.
• If the first counter is blue, there are $3$ red and $1$ blue counter left.

Start
├── Red 3/5
│   ├── Red  2/4
│   └── Blue 2/4
└── Blue 2/5
    ├── Red  3/4
    └── Blue 1/4

Each complete path shows one possible sequence:

• Red, Red
• Red, Blue
• Blue, Red
• Blue, Blue

Step 3: Find the probability of each successful path

“One red and one blue” can happen in two ways:

Red, then Blue

$$P(RB)=\frac{3}{5}\times\frac{2}{4} =\frac{6}{20} =\frac{3}{10}$$

Blue, then Red

$$P(BR)=\frac{2}{5}\times\frac{3}{4} =\frac{6}{20} =\frac{3}{10}$$

Step 4: Add the successful paths

The paths $RB$ and $BR$ cannot happen at the same time, so add their probabilities:

$$P(\text{one red and one blue}) =\frac{3}{10}+\frac{3}{10} =\frac{6}{10} =\boxed{\frac{3}{5}}$$

So, the probability of drawing one red and one blue is $\boxed{\frac{3}{5}}$.