There are 8 candidates for a position. The candidates are interviewed independen

Question

There are 8 candidates for a position. The candidates are interviewed independently by each member of a committee that has 4 people on it. A candidate will be hired if she is chosen by at least 3 members of the committee. In reality, the committee members cannot really distinguish among the candidates and just choose at random with no biases.

1- What is the probability that a candidate will be selected with a unanimous vote (all committee members choose the same person)?

2- What is the probability that a candidate will be selected with only one dissenting vote (all but one committee members choose the same person)?

3- What is the probability that a candidate will be selected?

PLEASE show me how to get them

Explanation / Answer

Every committee has an option to vote for 1 among 8 candidates.

So probability that a candidate will be voted for by a committee member = 1/8

1. The probability that a candidate will be selected with a unanimous vote by all 4 = 1/8 * 1/8 * 1/8 * 1/8

(as all voters are independent)

= 1/4096

2. Voter who will not vote for the candidate can be chosen in 4C1 ways = 4

What is the probability that a candidate will be selected with only one dissenting vote = 4 * (1/8)3 *(7/8) = 7/1024

3. Probabilty that a candidate will be selected = 1/4096 + 7/1024

as the possibilities in 1 and 2 are the only 2 possibilities to get selected

= 29/4096

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