I always loved staying home from school and watching Bob Barker, and now I love working from home and watching Drew Carey. But I always wondered what the best strategies for the wheel were.
The game is relatively simple. There are 20 different amounts in cents that can come up. Anywhere from 5¢ to $1.00 can come up. You get up to 2 spins to get more money on the wheel than the other two contestants, but if you go over $1.00, you “bust” and lose a chance to advance in the show. For now, lets ignore the problem of a potential tie.
So how should you approach this problem? Well, that depends on if you are the first, second, or third player. The strategy of the second and third spinners usually relies on beating the totals of the previous spinners, so lets focus on what to do if you are the first spinner.
There are 20 different possibilities on your first spin. You want to have a really high result, so that the other players will have a hard time beating your total. Well, how to solve that problem? Think about it this way:
The other players will have 2 chances to beat your spin. Let’s say your total was 75¢. That means that on their first spin, your opponents have 5 chances to better your total (80¢, 85¢, 90¢, 95¢, or $1.00). Which means they have a 5 out of 20 chance to win, or 25%. This seems logical, given that you spun a 75¢. But they get another spin!
Now here’s the tricky part, whatever they spin, they still have 5 chances to better your total! If they got 20¢, then they can better your total at 60¢, 65¢, 70¢, 75¢, and 80¢, but if they go any higher, they bust. If they got a 70¢, then they also have 5 chances; they could spin 10¢, 15¢, 20¢, 25¢, or 30¢.
But, really, you don’t care about their probability of winning, you care about your probability of winning, which is their probability of losing (100% -25% in this case) . Since each spin is a separate probability, we can simply multiple their losing probabilities together per spin. So for the second player, your chances of beating them are 75%*75% which is 56.25%. But wait! There’s a third player! His or her odds of beating your spin are the same as the second player, so to calculate the odds of beating both players, we again have to multiply our odds together. So 56.25% times 56.25% is now 31.60%! That’s a lot lower than the 75% we started with.
Of course, here’s the kicker. At 75¢, you could try to better your total with your second spin, but your odds of bettering your total is only 1 in 5, or 20%. So you have a 20% chance to better your total, or a 31.60% of winning if you sit on your total!
Now you could just assume that if your chance to better your spin is higher than your win chance, you should spin again, but you have to realize that successfully bettering your spin does not give you a 100% chance of winning! So you have to take into account the average improvement in your odds if you better your spin! For simplicity’s sake, I’ll just tell you the calculated value for your 75¢ spin. It’s 36.40% for a 75¢ spin. This means that if you spin the wheel a second time, and you don’t bust, on average, you’re improve your odds by 36.40%. If we multiply this improvement, by the probability, we get the value of a second spin (in this case, 20%*36.40% is 7.28%). Only if this value is higher than your win chance should you spin again! You can see on this chart where the values cross. At 65¢ and higher, you stand, 60¢ you spin again.
Now the next question is, do people really behave this way or not. That is a question for a future post from a psychology perspective.
