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This puzzle appeared in the 'Skeptical Adversaria' (the newsletter of ASKE, the Association for Skeptical Enquiry), December 2005.
In an earlier issue I commented that many interesting puzzles concern the number 3 (3 cards, 3 numbers, 3 people, 3 options, etc.). Probably the most celebrated of these, and deservedly so, is the Monty Hall problem. It is one of those puzzles for which intuition and logic (or, if you like, very simple mathematics) give different answers and, as is usual when they do, logic wins. That, of course, makes it of interest to skeptics. Another reason for this interest may be found in back copies of the Skeptical Inquirer. But for another account you can read The Curious Incident of the Dog in the Night-Time by Mark Haddon (Vintage 2004).
I am very grateful to ASKE member Jan Nienhuys from the Netherlands for assisting me with this task. Any existing errors are entirely due to me.
I expect that most, if not all, readers will be familiar with the problem and know its solution. If you do, please read on: there are some challenges ahead!
I prefer to present the problem in the form of the familiar 3 shells game of chance. I show you 3 shells, under one of which is a gold sovereign. You are to guess which shell (the ‘target shell’) covers the sovereign, and if you guess correctly, the sovereign is yours. However, the rules dictate that I must know which is the target shell and once you have made your choice, before you look under the shell, I must reveal to you one of the two remaining shells that does not cover the sovereign. Having done this, I must then offer you the chance of changing your mind, that is sticking with your original choice or switching to the shell that neither you nor I have indicated. What do you do – stick or switch?
If you do not know the answer, stop here while you work it out.
Before I give the answer, for those of you who are unfamiliar with this problem I shall express it in a form that makes the answer obvious.
Let’s label the shells A, B and C and suppose you choose shell A. Clearly you have a 1 in 3 chance of winning the sovereign. I then say to you, ‘If you wish, you can instead choose both shell B and shell C’. This raises your chance of winning to 2 in 3, so obviously you are going to accept my offer. But now I say, ‘Incidentally, don’t bother with shell B, I know the sovereign isn’t under there’. Obviously you then switch to shell C, still having a 2 in 3 chance that you will win the sovereign.
This variant on the earlier instructions clearly indicates that the answer to the problem in the original form is that you switch to the third shell every time and this doubles your chances of winning the sovereign.
I prefer the above explanation, but there are other ways of showing that switching is the correct strategy. One is to extend the number of shells to 100; once you choose a shell, out of the remaining 99 I turn over 98 that are empty. Obviously you switch to the remaining unselected shell.
For other proofs, enter ‘Monty Hall’ in Google and you will discover a host of websites. Or (unless you are averse to sentences that begin ‘And’) why not read The Curious Incident of the Dog in the Night-Time? Supposedly written by a very bright autistic boy, one of the strong impressions the story conveys is the selfishness and stupidity of the so-called ‘normal' adults in his life.
But there is more to the Monty Hall problem. Why is it that intuition strongly objects to the logical answer? One reason may be as follows. You already know that one or both of the two unselected shells must be empty. Hence it does not seem that I am giving you any extra information about the shell you have chosen by revealing to you one that is empty.
Let’s add to the intrigue by thinking about the situation in which before you make your choice I turn over an empty shell. For example, suppose that at the outset I reveal to you that shell B is empty. Have I given you useful information – i.e. information that will inform your choice of shell? Of course I have: now you only choose one of the two remaining shells. Let’s say you choose shell A, so your chance of winning has now increased from 1in 3 to 1 in 2. But if I reveal that shell B is empty after you choose shell A, your chance of winning with shell A is still only 1 in 3 but 2 in 3 with shell C. To put it another way, revealing an empty shell before you make your choice increases your chance of winning to 50%; revealing an empty shell after you have made your choice increases it by substantially more, namely to 67%, provided you switch!
Now, before you finally declare which is your chosen shell, any indication as to the identity of one of the empty shells must provide you with useful information that will affect your final choice. If this information is given before you make your choice, then you can only make good use of it by choosing one of the two other shells. But there is still uncertainty – the two shells are equally likely to cover the sovereign. So you still have a choice to make. If the information is given after your choice, the only way you can make good use of the information it is to switch to the third shell.
(Note: if I reveal an empty shell before you choose, I have a choice of 2 shells to identify. If I reveal an empty shell after you choose, in 2 out of 3 trials I have no choice as to which shell I reveal to be empty. Hence there is less uncertainty in the information I give you than if I indicate an empty shell before you choose.)
Suppose you are playing the Monty Hall version of the 3 shells game but before you make your preliminary choice I give you the following information. The shell allocated the sovereign has not been chosen purely at random (i.e. with a 1 in 3 probability) but so that the probability that it is shell A is 10%, shell B 40% and shell C 50%. Which shell should you choose first?
Suppose we play the Monty Hall game with more than 3 shells – 4, 10, 100, 1000, or whatever. There is still only one sovereign, and after you have made your preliminary choice, I reveal to you just one of the other shells that I know does not cover the sovereign. Do you still switch? If so can you work out a simple formula that expresses the advantage to you of switching over not switching when n shells are used?
Consider the case where there are just 3 shells. Suppose now that each time we play the game I have no idea under which of the shells the sovereign lies. I still turn over one of the two shells that you have not chosen. Obviously, sometimes I will turn over the shell that covers the sovereign, in which case you must lose. What should you do when I turn over an empty shell? Do you switch to the remaining shell or stick with your original choice?
The answers are given below, so do not look until you are ready.
The answers are as follows.
The answer is that you choose the shell with the lowest probability of containing the sovereign, namely shell A, with a 10% chance of winning. You thereby stand a 90% chance of winning because, once either shell B or shell C is revealed as empty, you can switch. Even though the initial chances of winning are greater with shell C (50%) than with shell B (40%) your chances of winning remain the same, at 90%, whether shell B or shell C is revealed as empty.
Note that if you were tempted to make shell C (50% likely to cover the sovereign) your initial choice, your chance of winning on switching would still only be 50%: switching conveys no advantage in this case, whether you’re left with shell A or shell B. If you initially choose shell B (40% likely to cover the sovereign) switching would convey a modest advantage (60% likelihood of winning whichever shell remains).
Let’s consider the case where there are 4 shells, A, B, C and D (and still only one sovereign). You choose one, say, A. Your chance of winning is 1 in 4, or 25%. The probability that the sovereign is under one of the other 3 shells is 3 in 4, or 75%. Say I turn over shell D, knowing it to be empty. The probability that the sovereign is under the remaining shells, B and C, stays at 3 in 4. But if you switch, you can only choose one of these, you can’t have them both. Whether you choose B or C, your chance of winning is half of 3 in 4, namely 3 in 8, or 37.5%, better than your original 25%.
So what is the general formula for n shells? The probability of winning with your initial choice is 1/n. The probability that any one of the other (n–1) shells covers the sovereign is: (n-1)/n.
When I eliminate one of these shells the probability that any one of the remaining shells covers the sovereign is still (n-1)/n.
There are (n–2) of these shells; therefore if you choose one of them your chance of winning is: (n-1)/n(n-2)
Clearly this reveals that the probability of winning when switching is always higher than when not switching (1/n) for any number of shells of 3 or more, (n–1) being greater than (n–2). Of course the advantage of switching diminishes with an increasing number of shells. For example, with 100 shells, your chance of winning if you do not switch is 1 in 10 or .01. If you switch, this rises to 99 divided by 100x98, which is .010102, hardly any advantage at all.
In fact, it can easily be shown that the increase in probability of winning by switching is: 1/n(n-2)
The answer is there is now no advantage in switching. At the outset, the probability that your chosen shell covers the sovereign is 1/3. The probability that my choice of shell covers the sovereign is also 1/3. (For random choices like this there is no advantage for whoever goes first.) Clearly the probability that the remaining shell covers the sovereign is also one 1/3. There is no way according to the present rules that you can increase your chance of winning. (Of course, if I choose an empty shell, the betting at that point is that your shell has a 1 in 2 probability of being the target shell-but so has the remaining shell. (Note that the probabilities are no different than when I turn over an empty shell before you make your choice.)
Another way of looking at this is to say that if my choice of shell is uninformed by my awareness of the identity of the target shell, then I must be conveying to you less information than when I turn over a shell that I know does not cover the sovereign.
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