Suppose n(>6) people are asked a question successively in a random order and exactly 3 out of n people know the answer. Let $p_{r}$ denote the probability that rth person asked is the first to know the answer, then

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Solution

The first person to answer the question will be the rth person if first (r-1) persons are chosen from (n-3) persons who do not know the answer and the rth person is chosen from 3 persons who know the answer. The probability of this event is
$p_{r} = \frac{^{n - 3} C_{r - 1}}{^{n} C_{r - 1}} \cdot \frac{3}{n - (r - 1)}$
A) As (n-3) people do not know the answer, probability that first four do not know the answer is
$\frac{^{n - 3} C_{4}}{^{n} C_{4}} = \frac{(n - 4) (n - 5) (n - 6)}{n (n - 1) (n - 2)}$
B) $p_{2} = (\frac{n - 3}{n}) (\frac{3}{n - 1}) = \frac{3 (n - 3)}{n (n - 1)}$
C) $∴ p_{n - 2} = \frac{1}{^{n} C_{3}}$
D) $p_{r} = \frac{^{n - 3} C_{r - 1}}{^{n} C_{r - 1}} \cdot \frac{3}{n - (r - 1)} = \frac{(n - 3)!}{(r - 1)! (n - r - 2)!} \cdot \frac{(r - 1)! (n - r + 1)!}{n!} \cdot \frac{3}{n - r + 1} = \frac{3 (n - r) (n - r - 1)}{n (n - 1) (n - 2)}$

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Suppose n(>6) people are asked a question successively in a random order and exactly 3 out of n people know the answer. Let pr denote the probability that rth person asked is the first to know the answer, then

Suppose n(>6) people are asked a question successively in a random order and exactly 3 out of n people know the answer. Let $p_{r}$ denote the probability that rth person asked is the first to know the answer, then