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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April 26th, 2003 at 9:29 am
Here’s the thing about price is right. I’d imagine that one should probably spin again on slightly higher numbers than the statistics indicate. The reason: The bonus $1000 since no matter what you spun on the first spin you always have a 5% chance of winning the big money. And since if you get the $1000 you get to spin again for either $10000 (5%) or $5000 (10%) that makes a successful 100 valued at $2000 this bonus would probably be enough incentive to skew the results slightly but the math is way to complicated for me since it involves comparing relative values of slightly diminishing the chance of making it into the showcase showdown with haveing a slight chance at a good size chunk of cash.
April 27th, 2003 at 6:54 pm
you misspelled “gory” as “gorry”
April 27th, 2003 at 8:54 pm
only could a UChicago student forever change the way i view a show in which i slurp down cereal in my boxers and scream at some contestant for underestimating the price of ALL laundry detergent.
God, i love my school.
November 24th, 2003 at 4:58 pm
Maybe I am missing something. But it seems that if there is a three way tie then the third person should spin again if that score is 65 or less as s/he will have a a 7/20 chance of winnng, slightly greater than 1 in 3.
And if there is a two way tie then s/he should spin agin if there is a tie at 45 or less as 11/20 is greater than 1/2.