Studio 33 - Price is Right Discussion > The TALK Is Right
The Ultimate Price is Right Strategy Guide
LiteBulb88:
Ack! Good catch--I do appreciate it. I can't edit my post here, but I have corrected my blog.
LiteBulb88:
Showcase Showdown (a.k.a The Wheel)
(Blog post: https://stoseontpir.blogspot.com/2019/06/the-ultimate-price-is-right-strategy_26.html)
Random factThe original wheel looked nothing like today's wheel; instead it looked like a carnival wheel with various colors. It's often referred to as the Rainbow Wheel. You can see the first playing of it here:
//www.youtube.com/watch?v=JeMJC1QFmgk
Who actually won? (seasons 29-46)
* The first spinner won 30.17% of the time.
* The second spinner won 33.81% of the time.
* The third spinner won 36.05% of the time, so a small, but not huge, advantage.Probabilities
(The following probabilities were calculated by enumerating all possible combinations of wheel spins and applying the strategy below.)Who wins:
* First spinner: 30.82%
* Second spinner: 32.96%
* Third spinner: 36.22%Getting $1 in one spin or a combination of two spins:
* At least one spinner: 23.03%
* At least two spinners: 2.35%
* All three spinners: 0.08%(Note the above three cases exclude the possibility of getting $1 in a spin-off, which would slightly increase the numbers.)
Having a spin-off with...:
* Exactly two players: 11.32%
* All three players: 0.48%StrategyThe only decision a contestant makes during this segment of the show is whether to take their second spin or not. How hard can that be? It turns out it's hard enough that people have written papers on it. So rather than attempt to derive the numbers here, I'll just spit out the results:
Spinner 1 strategy:
* Stay if you spun 0.70 or more.
* Spin again if you spun 0.65 or less.Spinner 2 strategy:
* If you spun less than spinner 1, spin again. (Duh.)
* If you spun more than spinner 1, or if spinner 1 went over, stay if and only if you spun 0.55 or more. If you are in the lead after your first spin but spun 0.50 or less, spin again.
* If you tied spinner 1, stay if and only if you spun 0.70 or more. If you tied spinner 1 but spun 0.65 or less, spin again.Spinner 3 strategy:
* If you spun less than the total of whoever is leading after the first two contestants have spun, spin again. (Duh.)
* If you spun more than the total of whoever is leading after the first two contestants have spun, stay. (Again, duh. In fact, Drew probably won't even let you spin again in this case.)
* If you have exactly one person to beat and you tied them, stay and take the spin-off if and only if you spun 0.55 or more. If the value you tied is 0.50 or less, spin again.
* If the first two spinners tied and you tied both of them, stay and take the spin-off if and only if you spun 0.70 or more. If you tied two people at 0.65 or less, spin again.
pannoni1:
--- Quote from: LiteBulb88 on June 26, 2019, 03:22:51 AM ---
Spinner 2 strategy:
* If you spun more than spinner 1, or if spinner 1 went over, stay if and only if you spun 0.55 or more. If you are in the lead after your first spin but spun 0.50 or less, spin again.
--- End quote ---
I'd certainly spin again if I spun .55 if there's just one spinner to go, since if you stay on 55 cents, although there's a 55% chance that you'll go over had you spun again, there would be a 45% chance that your opponent would beat you on the first spin, and if s/he would need a second spin, another 45% chance of accomplishing the task, putting the overall odds of the opponent winning at around 68%, and this doesn't take into account any spinoff(s). 60 cents is more even in terms of odds, meaning that there's a 60 percent chance you'll go over as well as about a 60 percent chance your opponent will beat you if you stay on 60 (40% on the first spin and another 40% on the second, but since a second spin is slightly more likely, this slightly tilts the odds in the "top winnners" favor) but since there's the possibility of money to be won by getting a dollar, I'd also spin again.
LiteBulb88:
The problem with that analysis is that this is the equation it is trying to satisfy:
P(not going over if you spin again) > P(winning if you stay)
Those are not the correct two quantities to compare. It's an easy mistake to make. Instead, you need to compare these two quantities:
* P(winning if you spin again)
* P(winning if you stay)
P(winning if you spin again) is NOT the same as P(not going over if you spin again), as there spins that don't cause you to go over but don't help you win either. So let's look at these one at a time for the case where spinner 2 spun 55 cents on the first spin and that score strictly beats the first spinner (or the first spinner went over):
P(winning if you stay)
There are two ways to win if you stay:
* Spinner 3 strictly loses to you
* Spinner 3 ties you and you win the spinoff.
Let's look at those one at a time.
P(spinner 3 strictly loses to you)
For spinner 3 to lose to you, they must spin less than or equal to 50 cents on the first spin--probability 10/20--and then on their second spin, spin a number that does NOT result in them having a total of 55, 60, 65, ... 100. That's also a probability of 10/20. Thus, the probability that spinner 3 strictly loses to you is 10/20 * 10/20 = 1/4.
P(Spinner 3 ties you and you win the spinoff)
Spinner 3 has two ways to tie you: spin 55 cents on the first spin (probability 1/20) or spin 55 in a combination of two spins. The latter case requires them to spin 50 or fewer cents on the first spin (probability 10/20) and then spin the exact amount to get to 55 cents on the second spin (1/20). The probability of spinner 2 winning the spinoff is 1/2, so the total probability here is:
P(winning the spinoff) * (P(spinner 3 tying you in one spin) + P(spinner 3 tying you in two spins))
= 1/2 * (1/20 + 10/20*1/20)
= 3/80
Thus, the total probability of spinner 2 winning if they stay on 55 cents is P (strictly beating) + P (tying & winning the spinoff) = 1/4 + 3/80 = 23/80 = 28.75%
P(winning if you spin again)
If you spin again, you have a 1/20 chance of ending up with 60 cents, a 1/20 chance of ending up with 65 cents, ..., a 1/20 chance of ending up with 100 cents (a dollar), and an 11/20 chance that you go over. Thus, your probability of winning if you spin again is:
1/20*P(winning with a total of 60 cents) + 1/20*P(winning with a total of 65 cents) + ... 1/20*P(winning with a total of 100 cents) + 11/20 * P(winning if you go over).
That last term is, of course, 0, so you can throw it out. So we need to calculate the probability of winning with 60 cents, 65 cents, etc. Well, we just did that for the 55 cents case above, so you can apply the same logic with slightly different numbers. I won't bore you with all the math (I did it in an Excel spreadsheet while working on this article), and instead tell you the result of the sum above is about 0.28031, or 28.031%. So your probability of winning decreases if you spin again on 55 cents, and thus you should not spin again.
In fact, here are all the numbers in case you're curious. Again, the below chart assumes that spinner 2's first spin strictly beats spinner 1 or that spinner 1 went over. I've bolded the rows where it changes from "they should spin again" to "they should not spin again."
First spin P(spinner 2 wins if they stay) P(spinner 2 wins if they spin again)
5 0.250% 33.988%
10 0.875% 33.944%
15 2.000% 33.844%
20 3.625% 33.663%
25 5.750% 33.375%
30 8.375% 32.956%
35 11.500% 32.381%
40 15.125% 31.625%
45 19.250% 30.663%
50 23.875% 29.469%
55 28.750% 28.031%
60 34.125% 26.325%
65 40.000% 24.325%
70 46.375% 22.006%
75 53.250% 19.344%
80 60.625% 16.313%
85 68.500% 12.888%
90 76.875% 9.044%
95 85.750% 4.756%
100 95.125% 0.000%
LiteBulb88:
--- Quote from: pannoni1 on June 26, 2019, 07:42:23 AM ---but since there's the possibility of money to be won by getting a dollar, I'd also spin again.
--- End quote ---
Forgot to respond to this. The average amount of money you'd expect to win if you spin 55 cents on your first spin and spin again is:
P(winning exactly $1,000) * $1,000 + P(winning exactly $11,000) * $11,000 + P(winning exactly $26,000) * $26,000.
Winning exactly $1,000 requires two things--first, you spin 45 cents (probability 1/20) and then you do NOT hit a bonus value on the second spin (probability 17/20). Total probability: 1/20*17/20 = 17/400
Winning exactly $11,000 requires you to spin 45 cents and then you spin 5 or 15 cents. Total probability: 1/20 * 2/20 = 1/200
Winning exactly $26,000 requires you to spin 45 cents then 1 dollar in that order. Total probability: 1/20*1/20 = 1/400.
Thus, the equation above becomes:
17/400 * $1,000 + 1/200 * $11,000 + 1/400 * $26,000 = $162.50
I personally wouldn't be willing to decrease my odds of going to the showcase in order to win, on average, $162.50.
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