Question

A volunteer from a group of three individuals is be chosen by having each toss a coin with
probability of head α. 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)

Answer 1

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