Imagine a group of contestants at a TV quiz show. Behind a screen there is an urn containing 10 balls which can be either Red or Blue. The group must guess whether the majority of the balls is Red (alternative R) or Blue (alternative B). There are two scenarios: either there are 6 Red balls and 4 Blue balls, or the reverse. Every member of the group, alone and without being seen by the others, goes behind the screen, extracts a ball from the urn, sees its color, puts the ball back, and casts a vote for R or B. If the majority of the votes are correct, every member of the group wins a prize (the same prize).
- 1.We can think of the ball each individual extracts as an independent signal. What is the probability that the signal is correct?
- 2.Suppose the group has a single member. How is he going to vote? What is the probability that he wins the prize?
- 3.Suppose the group has 3 members? What is the probability that the group wins the prize? Suppose the group had 5 members – can you write the equation that determines the probability that the groups wins the prize in this case?
- 4. Suppose the television audience is asked to play – by calling a number you receive a computer-generated text message on your cell phone with the color of the ball you would have extracted, and can reply with your vote. 50000 people call in. What is the probability that a majority of the audience guesses correctly?
- 5.Suppose a member of the audience learns all voting rules are subject to strategic manipulation. Call sincere voting here always voting according to one’s signal. Can the student gain by voting non sincerely? Why or why not?