Problem #2 (20 points) -Every Vote Counts Imagine that the United States Senate

Question

Problem #2 (20 points) -Every Vote Counts Imagine that the United States Senate is about to vote on an important bill Let's suppose that there are A senators (out of N) that are "against" the bill (with A N/2 senators "for" the bill so that if all N senators show up for the vote, the bill will past. However, suppose that M of the senators miss the vote because they are not present during the day of the vote and suppose that their absence is due to purely random reasons (traffic delays, illness, forgetfulness, etc.) and has nothing to do with whether a senator is either for or against the bill. It is then possible for the bill to go the wrong way, i.e., for the senators who do show up to vote to collectively defeat the bill a.) (15 points) In this problem, you want to compute the probability that the bill is defeated given that M of the N senators miss the vote. You may leave your final answer as a sum but clearly explain the terms in the sum and the limits in the summation. Hint: Let X be the random variable on the number of senators missing the vote that, if present, would have voted for the bill. Then compute P(X = x|A, F, M) and go from there b.) (5 points) Compute the probability that the bill is defeated given that 49 M 10 of the N and F 51. 100 senators miss the vote assuming also that A

Explanation / Answer

PART a)

Total number of Senators = N

Number of senators against the Bill = A ( < N/2 )

Number of senators for the Bill = F = N - A ( > N/2 )

Number of absent Senators = M

Lets say that, Number of absent Senators for the Bill = X

Number of absent Senators against the Bill = M - X

=> Number of present Sentors against the Bill = Anew = [A - (M - X)]

=> Number of present Senators for the Bill = Fnew = [F - X]

So, the probability that the Bill is defeated = P(Anew > Fnew)

= P( [A - (M - X)] > [F - X] )

= P( [A - M + X] > [N - A - X] ) = P( X > (N+M)/2 - A ); Given the constraint: X <= M (always)

= Probability of [ M <= X < (N+M)/2 - A ]

= Sum of [ Probability(X = x)]...... for x = ceiling[(M+N)/2 - A] to x = M (all discrete values)

where, P(i) = 1/(M+1) (because, X being number of Senators absent from the 'For' side, is a random variable which can take any discrete value from 0 to M with uniform probability)

=> Answer = [1/(M+1)]*(Number of integers between ceiling[(M+N)/2 - A] and M) ... (only discrete values)

PART b)

For N = 100, F = 51, A = 49, M = 10...

lower bound = ceiling[(M+N)/2 - A] = ceiling(6) = 6

upper bound = M = 10

=> Number of discrete values available = 5

=> Probability of Bill being defeated = 5*(1/11) = 5/11[Answer b.]

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