A tree diagram or probability tree can help to solve probability problems or problems involving the number of ways that a combination of things can be carried out.

Tree diagrams are useful for solving probability problems with more than one stage.

The probabilities on each set of "branches" always add up to 1. They are mutually exclusive events.



Probability Rules

On a probability tree:

to find the probability of event A and event B happening multiply probabilities across the tree.

to find the probability of event C or event D happening add probabilities down the tree.

Example 1

A rugby statistician has worked out that the probability that goal kicker in a rugby game kicks a penalty successfully if it is windy is 0.6 and if it is not windy is 0.7.
The probability that the wind will blow when he takes the kick is 0.2.
Draw a probability tree to show this situation and find the probability that the kicker is sucessful with the penalty.


P(windy and kicks penalty) = 0.2 x 0.6 = 0.12


P(not windy and kicks penalty) = 0.8 x 0.7= 0.56


P(kicks penalty) = P(windy and kicks penalty) or
P(not windy and kicks penalty)

                           = 0.12 + 0.56

                           = 0.68



Probability trees are especially useful for problems involving the probabilities of events happening where items are selected without replacement.

Example 2

Ten coloured balls are placed in a bag. Six of the balls are red and four of the balls are black. Three balls are drawn out of the bag and as each one is drawn it is not replaced.

Draw a tree diagrams to show this situation and find the probability that all three balls are the same colour.


P(all three are same colour)

= P(red and red and red) + P(black and blackand black)

= P(R Y12_Tree_Diagrams_04.gif R Y12_Tree_Diagrams_04.gif R) + P(B Y12_Tree_Diagrams_04.gif B Y12_Tree_Diagrams_04.gif B)