Probability Tree Diagrams

Probability Tree Diagrams


Probability can be presented using tree diagrams. Each branch of the tree represents an outcome (similar to a frequency tree diagram, but each branch is labelled with a probability, not a frequency).

All outcomes must be shown from each node. For example, a bag of balls contains 4 red balls and 6 blue balls. P(red) = `frac(2)(5)` and P(blue) = `frac(3)(5)`:

If, in the above example, a ball is drawn, replaced and then a second ball is drawn:

Multiply the probabilities along branches to calculate the probability of two consecutive events. The probability of drawing two red balls is P(red) x P(red) = `frac(2)(5)` x `frac(2)(5)` = `frac(4)(25)`.

Adding the end result each of the routes along the branches gives a probability of 1.


1. A bag contains 12 balls. 3 are red and 9 are blue. A ball is drawn at random, and replaced in the bag. A second ball is drawn at random. What is the probability of drawing 2 red balls?

Answer: `frac(1)(16)`

Draw the probability tree. For two red balls, multiply along the tree:

P(red) x P(red) = `frac(3)(12)` x `frac(3)(12)` = `frac(9)(144)` = `frac(1)(16)`

2. The experiment is repeated with the same bag. In this instance, the balls are not replaced. By drawing a probability tree, or otherwise, show that the probability of drawing two consecutive balls of the same colour is `frac(21)(33)`.

Answer: Probability of two reds or two blues (note balls are not replaced):

(P(red1) x P(red)2) + (P(blue)1 x P(blue)2)

`frac(3)(12)` x `frac(2)(11)` + `frac(9)(12)` x `frac(8)(11)` = `frac(13)(22)`