## Wednesday, December 9, 2009

### The Price Is Right Wheel

If you're ever on The Price is Right, and you get to spin the big wheel, and someone gets to spin the wheel after you, what spin should you use as the cut-off for respinning? Now you may say, "anything over 50 means I'm likely to be highest, right?" Let's figure it out. The next spinner has a 100-your_spin chance of beating you in one spin, plus a your_spin * (100-your_spin) chance of beating you in two spins. If you spin a 70, the next person has a 30 + 70*30 percent chance of beating you, or 51%. Of course, you have a your_spin% chance of going bust if you spin again, and a greater chance of losing if two people spin after you than one.

So, if you spin a 60 as the first spinner, your chance of losing if you keep it is ((second spinner 40% + 24%) + (third spinner if second fails 36% * 64%)) = 87%. As the second spinner (if you beat the first), you have a 64% chance of losing. These are both higher than break-even (50%), and your chance of going bust on an additional spin is 60%, which is less than the chance of losing if you keep the 60. Clearly, if you spin a 60, you should always spin again. The stay point is higher.

If you spin a 65, your chance of losing as the first spinner is (35% + 23%) + (58% * 42%) = 82%, and 58% as the second spinner. In this case, the chance of blowing a second spin is 65%, so a first spinner should spin again on a 65, but a second spinner should stay on a 65 if it beats the first spinner.

With a 70, a second spinner has a 51% chance of losing, and a first spinner has a 51% + (51% * 49%) = 76% chance of losing. The chance of blowing another spin is again in the middle.

It is at the 75 point that a second spinner has a 44% chance of losing, and a first spinner has a 44 + (44*56) = 69% chance, but the chance of a fatal second spin is 75%. So, no one should spin again on a 75.

Summary:
First spinner should always spin again on anything less than 75.
Second spinner should always spin again on anything less than 65.