More nerdy statistics

Here’s this week’s take-home lesson from Statistics 251, Introduction to Mathmatical Probability:

We’re all familiar with the gameshow The Price Is Right. In part of the show, three contests spin a wheel numbered from 5-100. The goal is to get as close to 100 as possible within 2 spins without going over. Obviously, if your first roll is low, you should roll again and try to improve the score. But if your score is high, you shouldn’t roll again because you’ll probably go over 100. Note that in the case of a two-way or three-way tie, the contestants have a tie breaker in which each contestant has an equal chance to win.

So the question is, what is each contestant’s optimal strategy? (For what scores should contests roll again or pass to maximize their chances of winning?) And, what is each contests chance of winning if all contestants follow the optimal strategy?

I’ll spare you the gory details and just tell you the answers, since you’re not taking the class.

The first contestant should spin again if his/her score is a 65 or less.

If the first contestant spun a score j, the second contestant should spin again if his/her score is 10 and j <= 10, or if the first spin is j and 11 <= j <= 13, or if the first spin is j-1 and 14 <= j <= 20.

If the third contest spins a score k that ties the current score, s/he should spin again if k <= 10 and there's a two-way tie, or k <= 13 and there’s a three-way tie.

If every player follows the optimal strategy, the contestant’s respective chances of winning are 30.8%, 33.0%, and 36.2%. Thus, going last has a moderate advantage.

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