New Monty Hall Problem?

Question

You're at a cheap carnival. A sharp operator accosts you, and convinces you to try a game with cups and a pea. You agree. He shows you 5 cups upsidedown, one of them covering a pea, and asks you if you can guess which one? You choose one. He looks at you for a moment, and then decides to turn over 3 other cups, showing no pea, so that either the remaining cup has the pea, or yours does. He says you can change your mind if you want. What should you do?

For those not familiar with the Monty Hall Problem, check wiki article on it:

http://en.wikipedia.org/wiki/Monty_Hall_problem

Answer 1

Difficult to say, since we don't know under what circumstances he would turn over the other cups, nor do we know whether he would avoid turning over the cup with the pea, if it was different from the one I had selected. Why is this relevant you ask? Isn't the probability that I chose correctly 1/5 regardless of what he does now? To answer that question, consider the situation where instead of turning over three empty cups, he turns over all four empty cups and asks whether I'd like to switch to a different cup. Obviously I would be a fool to do so, since if all of the other cups are empty, I MUST have chosen the correct cup (assuming he didn't simply palm the pea while he was shuffling). By turning over all four of the remaining cups, he has provided me with information about the correctness of my original choice. So the relevant probability here is not the probability that I chose correctly initially, but the probability that I chose correctly given that he has turned over three cups, showing no pea.

Now, in the original Monty hall problem, the probability that you chose correctly in the beginning, and the probability that you chose correctly in the beginning given that he has just opened a door showing a goat, are the same, since he ALWAYS opens a door showing a goat. To see why this is important, consider a modified problem where he opens a door other than the one you selected at random, whether it contains a car or a goat. In this case, we have:

IC = probability that your initial choice was a car
IG = probability that you initial choice was a goat
MC = probability that Monty opens a door hiding the car (note this would never happen in the original problem)
MG = probability that Monty opens a door hiding a goat (note that this would always happen in the original problem)

P(IC) = 1/3, P(IG) = 2/3
P(MC | IC) = 0 (he can't open a door containing a car if you've already picked that door)
P(MG | IC) = 1
P(MC | IG) = 1/2, P(MG | IG) = 1/2
P(MG) = P(MG | IC)*P(IC) + P(MG | IG)*P(IG) = 1*1/3 + 1/2*2/3 = 2/3
P(IC | MG) = P(MG | IC) * P(IC) / P(MG) = 1*1/3 / (2/3) = 1/2
P(IG | MG) = P(MG | IG) * P(IG) / P(MG) = 1/2*2/3 / (2/3) = 1/2
P(IC | MC) = 0 (duh)

In other words, while the probability that you chose correctly in the beginning remains the same, if Monty is known to choose randomly rather than always choosing a door hiding a goat, then the fact that he did not show you the door containing the car (and thus deprive you of any chance of winning whether or not you switch) gives you information -- information that suggests you chose correctly in the beginning. As such, the odds that you will benefit by switching are lower than if you had no information about the correctness of your original choice (which was the case in the original monty hall problem).

So going back to the original problem with the pea and the cups, I need to know whether the fact that he has revealed three empty cups tells me anything about the correctness of my original choice. If he ALWAYS turns over three empty cups, then this obviously gives me no information about my original choice whatsoever, which means that I more than likely chose wrong and should switch to the remaining cup. If he always turns over three cups other than the one you selected, but does so randomly (and thus sometimes reveals the pea in the process), then the fact that he did not reveal the pea this time tells me that I'm more likely to have been correct the first time -- in fact, in such a case, the probability that I would gain by switching is precisely 50%. Finally, if he only offers the choice to switch when you actually picked the right cup the first time (which given that I'm at a cheap carnival, seems far more likely), then the fact that he gave me the chance to switch tells me with certainty that my original choice was correct, and so I should never switch.

Answer 2

Just as in the original Monty Hall problem, it is advantageous to switch to the other cup. However, the advantage is even greater in this case.

In the original problem, you had a 1/3 chance of winning if you stuck with your first choice, and a 2/3 chance of winning if you switched.

In this problem, you have a 1/5 chance of winning if you stick with your first choice, and a 4/5 chance of winning if you switch.

Update:
Anyone who suggests the probability of winning is 1/2 has no idea what they're talking about. It simply isn't. Consider a million cups and one pea. You choose a cup. The chance you've chosen the right cup are 1 in a million. Very slim odds. Now the operator removes 999,998 cups that don't contain a pea. There are two left. If you stick with your original choice, the odds of winning CANNOT DEVIATE from 1 in a million, since no event occurred that could affect the outcome. Therefore, the probability of winning if you switch is 999,999/1,000,000.

Answer 3

Yes.. Change your mind.. If you stay with the first cup, your probability is 1/5 of getting the pea. Once he shows you three cups without the pea, your probability increases if you change your mind. If you stay, then you havent taken advantage of the new information, so your probability is still 1/5.

It has to do with what information is known at the beginning of your choice.

Answer 4

it doesn't matter...

in the original problem, you had 1/5 chance of picking the right one...

now that he has uncovered 3, you only have a 1/2 chance of picking the right one...

so either way you go, you have 1/2 chance of being right...

Answer 5

I would stick to my first choice. Gut instinct is always good. It is never a good option or judgment to second guess yourself. that is my opinion.