A volunteer from a group of three individuals is be chosen by having each toss a

Question

A volunteer from a group of three individuals is be chosen by having each toss a coin with
probability of head . The individual with the unique outcome is chosen. If the three
tosses result in the same outcome, then the three individuals toss their coins again, and the
process continues until a volunteer is chosen.

a) Let q denote the probability of a unique outcome in a given trial. Find q

b) Let N denote the number of trials it takes to find a volunteer. Find the pmf pN (n).

c) Let P(N < k) denote the probability that a volunteer is found in less than k trials.
Find an explicit expression for P(N < k)

Explanation / Answer

here head probability = a

tails probability = 1- a

so here

(a) q = Pr(Unique outcome) = 1 - Pr(same outcome) = 1 - [Pr(All heads) + Pr(all tails)]

= 1 - [a * a * a + (1 - a) * (1-a) * (1-a)]

= 1 - [a3 + (1 - a)3]

= 1 - a3 - (1 - a)3

= 1 - a3 - (1 - 3a + 3a2 - a3 )

= 1- a3 - 1 + 3a - 3a2 + a3 = 3a (1 - a)

(b) Here N is the number of trials it takes to find a volunteer

so here the distributioon of n would be geometric distribution with probability of success = 3a(1-a)

so here

p(n) = Pr(failure in n-1 chances) * Pr(success in nth chance)

= (1 - 3a + 3a2)n-1 * 3a (1 - a)

(C) Here that volunteer is found in less than k trials.

P(n) = P(N < k) = 1 - Pr(There is no success in k-1 attempt) = 1 - (1 - 3a + 3a2)k-1

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