By Andrew Bernoff, professor of mathematics

In the popular game show “Who Wants To Be a Millionaire?” contestants are asked trivia questions with four possible answers (labeled A,B,C and D). If the contestant does not know the correct answer, one of the things he or she can do is ask the audience members to vote for what they think the correct answer is. Conventional wisdom holds that the audience is almost always right.

Suppose there are 100 people in the audience and we assume that each one either votes for the correct answer (if they know it) or chooses randomly one of the four responses with equal probability (if they do not know the answer).

CHALLENGE:
How many votes must the most popular answer to the question receive so that the probability that the answer is correct is at least 50%? 90%? 99%?

Suppose that 10 members of the audience know the answer. What is the probability that the correct answer will receive the most votes? What if only five people know the correct answer?