Lesson 8

In this problem, the probability space is again given implicitly. And again, finding its explicit representation will be helped by the process tree—it is shown in the figure below. The number we are interested in in this tree is the proportion of highlighted events in the left subtree (where the first team won the first game): \[\frac{1/3+1/18}{1/3+1/18+1/9}=\frac{7}{9}\ .\] This same number equals the proportion of the shaded part in the left half of the square in the figure.

Left: process tree of the previous problem. Letters W and L denote the win and loss of the first team, respectively. The outcomes in which the first team wins the tournament are highlighted. On the right, the space of elementary outcomes is shown: we start with a unit square and at each subsequent game divide the current rectangle in the corresponding proportion.