There are 100 US senators two from each of the 50 states a i

There are 100 US senators, two from each of the 50 states.

a. if 2 senators are chosen at random for a fact-finding mission, what is the probability that they are from the same state?

b. If 2 senators are chosen at random for a fact-finding mision, what is the probability that they are from Masssachusetts?

c. If 10 senators are randomly chosen to form a committee, what is the probability that they are from different states?

Solution

There are 100C2 = 4950 ways to choose 2 senators.

a)

There are 50 ways to choose which state.

Thus,

P(same state) = 50/4950 = 0.01010101 [answer]

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b)

There\'s only 1 way to do this.

P(both massachusetts) = 1/4950 = 0.00020202 [answer]

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c)

There are 100C10 = 1.73103*10^13 ways to choose 10 senators.

There are 50C10 ways to select 10 differnet states, and 2^10 ways to choose which senators represent those 10 states.

hence,

P = [(50C10)*2^10] / [100C10]

P = 0.607661745 [ANSWER]