Chapter 15.3, Problem 2P

Prove (3.1) for a nonuniform sample space. Hints: Remember that the probability of an event is the sum of the probabilities of the sample points favorable to it. Using Figure 3.1. let the points in $A$ but not in AB have probabilities $p_1,p_2,\dots ,p_n,$ the points in AB have probabilities $p_{n+1},p_{n+2},\dots ,p_{n+k},$ and the points in $B$ but not in AB have probabilities $p_{n+k+1},p_{n+k+2},\dots ,p_{n+k+1}.$ Find each of the probabilities in (3.1) in terms of the $p$ ’s and show that you then have an identity.