Tree diagrams for other random situations

We introduced tree diagrams on the previous page for two events that we considered to happen in sequence.

Tree diagrams can be used for any situation involving a sequence of events. Each possible sequence of events is represented by a path through the tree diagram. Branches are labelled with the probability for the branching conditional on the branches to the left.

The joint probability for any sequence of events is again found by multiplying the probabilities on all branches of the tree corresponding to this sequence.

The ideas are more clearly explained in an example.

Boys and girls in a family

A couple plan their family as follows. They want at least two children and no more than four. However, subject to this constraint on their total number of children, they will stop when they get a boy.

Assuming that there are no multiple births and the probability of any child being male is 1/2, independent of the genders of previous children,

The following tree diagram represents the possible sequences of births.

In this tree diagram, all branches have conditional probability 1/2 so they are not displayed. The joint probabilities at the end of the branches are found by multiplying these 1/2's together.

Since the different branches in a tree diagram are mutually exclusive, their probabilities can be added to find the required probabilities:

Family size
The probability that the family ends up with two children is the sum of the probabilities for the sequences BB, BG and GB,

P(family size = 2)   =   0.25 + 0.25 + 0.25   =   0.75

The other probabilities are shown in the following table.
Family size, x 2 3 4
P(X = x) 0.75 0.125 0.125
Number of girls
Adding branch probabilities corresponding to the different numbers of possible girls in the family gives the following probabilities:
Number of girls, y 0 1 2 3 4
P(Y = y) 0.25 0.5 0.125 0.0625 0.0625