What Are The Odds?: The Big Wheel

Saturday, December 22, 2007

Have gotten some e-mails on this subject, so figured I’d share some juicy mathematical (if not common sensical) knowledge.

Assuming the Big Wheel is random and you can’t control where your spin lands– at what amounts do you stay, and what amounts do you spin again? – Various readers

Here’s the easy-to-follow cheat sheet:

Player 1: Spin again if your first spin is .65 or less. The reason for this being is that for both of the remaining spinners to lose to you, they need to spin less than .65 on their first spin, and then more than .35 on their second spin– this will happen 33% of the time. At .65, the odds of you spinning between .05-.35 are greater than (albeit slightly) 33%, so this is the breakeven value.

Player 2: Spin again if A) you have less than Player 1 (Duh!), B) you spun .50 or less regardless of what player 1 spun, or C) you have tied player 1, but have .65 or less. A is self-explanatory. You can’t win finishing 2nd place. Choice B just means that the third spinner has a 50/50 chance of finishing higher than .50 (counting one spin, or a combination of 2). Therefore, if you are at .40, say– you have a 60% chance of improving your score without going over. If you stay, the odds are greater than 50% than the third spinner will pass you, meaning you should spin again and try to decrease their odds of beating you. Choice C means while you might spin into a tie, you need to consider the third spinner’s prospects for beating you. For example, if you tie at .65, the third spinner has a 35% chance of beating both of you. Now, while 33% < 50%, (the aforementioned break-even point for spinner #2) you also have to consider the prospect of a spinoff, which you are still 50% to win, even if the third spinner still fails to top you're current score. So at the magical .65 value in a tie, you are 33% to win, and have a 33% chance to spin higher. Hence the break-even point.

Player 3: Spin again if A) you are losing (another duh!), B) if you are tied with one person at .45 of less, or C) in a three-way tie at .65 or less. Point B means that you have a 50/50 chance of winning a spin off, but for every tie you have less than .50, you have a greater than 50% chance to improve your score (and break the tie) so you might as well take your chances spinning again. Point B essentially says the same thing, but in this case you’re only a 1 in 3 chance to win a random spinoff, so if you are in a three-way tie at less than .65, you might as well spin again since the odds of you breaking the tie without going over are > 33%.

So for the Cliffnotes version:

Spinner 1:
- Spin again if you are .65 or less.

Spinner 2:
- Spin again if you are losing to spinner 1.
- Spin again if you beat spinner 1, but are still .50 or less.
- Spin again if you are tied with spinner 1, but are .65 or less.

Spinner 3:
- Spin again if you are losing to either previous spinner
- Spin again if you are tied with one spinner at .50 or less.
- Spin again if you you are in a three-way tie at .65 or less.

Good luck!

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