A tree diagram visually represents possible outcomes of multiple-step probabilistic events. Each branch splits to represent different outcomes of each stage, with probabilities written along the branches. To find the probability of a particular combination of events, multiply probabilities along its branches. To find the probability of an event with multiple outcomes, sum the probabilities of each relevant branch. Tree diagrams simplify complex probability problems, clearly showing how probabilities combine at each step.

Example

Marbles are drawn from a bag containing 7 red marbles and 5 green marbles. Jack draws 2 marbles from the bag. Find the probability that he selects

• two red marbles

• two marbles of different colors.

We fill in the probabilities on the first branches in the tree. The probability that the first marble is green is 125​, and the probability that it is red is 127​.

After drawing the first marble, there are 11 left. There is also one fewer red / green marbles. So the probability of drawing a red after drawing a red is 117−1​=116​.

The probability of drawing two red marbles is thus

The probability of drawing two marbles of different colors is