A probability tree diagram expresses the possible outcomes of a given scenario. Math instructors often ask students to construct tree diagrams in order to solve probability problems. Students working with tree diagrams for the first time should begin by understanding express the probabilities associated with a scenario that has two possible outcomes. Examples include finding the probability of randomly selecting a marble from a bag containing two colors or the probability of getting either heads or tails when flipping a coin.

## Instructions

1. Draw a large “greater than” sign, <, which represents the first two branches of the tree. Each branch represents the outcome of a scenario.

2. Suppose that you have a bag containing 12 red marbles and eight white ones.

3. Place a dot on the point where the two branches meet. The dot represents the first event, whose probability is the sum of the probabilities assigned to its branches.

4. Indicate which branch represents each scenario. Write “Red,” or simply “R,” next to one branch and “White,” or “W,” by the other branch.

5. Write the probability of each scenario occurring, i.e., the probability of selecting a red marble from the bag. There are 20 marbles in total (8 white + 12 red), so the probability of picking a red one is 12/20. Write 8/20 next to the second branch. You can also express each probability as a percentage, but expressing it as a fraction will facilitate calculations that you may have to perform later on.

6. Represent the probability of picking another red or white marble by expanding the tree diagram. Draw another “greater than” sign, connected by a dot, coming off of each end of the original branches. You will now have four new branches on the tree.

7. Use the same system to label the first two branches to represent the scenario of selecting another red or white marble after a red one has been selected. Similarly, label the two remaining branches to represent the scenario of selecting another red or white marble after a white one has been selected. Because you removed one of the marbles during the previous round, express the probabilities in the second round of picking marbles from the bag out of 19, not 20.

8. Keep adding branches and the corresponding probabilities if the problem in question involves more scenarios.

9. Multiply the probabilities of multiple branches to determine the likelihood of a specific sequence of events. Suppose that you must find the probability of selecting two red marbles in a row. The probability of selecting a red marble during the first around is 12/20. During the second round, the probability would be 11/19 because there are 19 marbles left in total and 11 red ones. Therefore, the likelihood of picking a red marble and then another red one would equal to the product of 12/20 and 11/19, or 132/380.