Probability of state representation

Question

Just working my way through the W.Feller introduction to probability theory and its applications volume 1. An example in the chapter on combinatorial analysis asks the question:

"Each of the 50 states has 2 senators. If we choose 50 senators at random, what is the probability a given state is represented?"

I understand the answer given which uses the complement of the event but was curious whether the method where you force the desired outcome to occur, then work out how many ways the remaining cells can be chosen, would work here too?

AJ

Answer 1

Let s1 and s2 the two senators of the state.

P(state is represented) = P(s1 or s2 is chosen by chance).

Let us compute the respective numbers of favorable cases:

• s1 and s2 chosen: 48 to choose in the 98 remaining senators
• s1 chosen without s2: 49 to choose in the 98 remaining senators
• s2 chosen without s1: the same
• none of them chosen: 50 to choose in the 98 remaining senators

that is, P(state is represented) = (98!/48!50! + 2*98!/49!49!) / (98!/48!50! + 2*98!/49!49! + 98!/48!50!)