Question

Each state of the 50 in the United States has two senators. What is the probability that in a committee of 50 senators (a) Iowa is represented, (b) all states are represented?

Answer 1

Since each state has 2 senators, so we have in total 100 senators. We have to choose a committe of 50 out of these 100 senators.

(a)

Probability that a state is represented = 1 - P(that state is not represented)

Now, a state will not be represented if both its senators are not selected. This means that 50 member committe is selected from remaining 98 senators excluding the senators of that state.

So,

P(a state is not represented) = ⁹⁸C₅₀/¹⁰⁰C₅₀ = 0.247

So,

P(a state is represented) = 1-0.247 = 0.753

So,

P(Iowa is represented) = 0.753

(b)

In this case we want each state to have exactly one of its senator present in the committee.

Each state has two options, and there are total 50 states, so there are total 2^50 possible ways of choosing a 50 member committe so that one member is selected from each state.

So,

P(all states are represented) = (2^50)/(¹⁰⁰C₅₀) = 1.12*10^-14