Thread: Find the Coin puzzle
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Old 06-22-2010, 09:43 AM   #18
pdurrant
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OK, answers time. I've put them in spoiler tags for anyone late to the puzzle.
If you're still confused after readin these spoilers, see the wikipedia article on the Monty Hall Problem, which is very similar.


Quote:
Originally Posted by pdurrant View Post
Question 1: On average, are you going to be better off moving your bet and doubling it, staying put, or does it not make any difference which you do?
Spoiler:
On average, you're better off doubling your bet and switching. That turns a 1/3 coin loss per game into break even.

Here's why.

You have a one in three chance of picking the right cylinder initially. The showman can always up a cylinder that doesn't have a coin under it. Because he can always do it, it doesn't affect your odds. After he's lifted a cylinder, your cylinder still has the one in three chance of being right. The other remaining cylinder has a two in three chance of being right.

Another way to look at it is this. You've picked a cylinder. The showman offers you the chance to switch to choosing BOTH of the other cylinders instead. This is obviously switching from a 1 in 3 to a 2 in three chance. Whether he shows you before or after switching that one of the other two cylinders doesn't hide the coin is irrelevant - you already know that at least one of the cylinders doesn't hide the coin.

So if you stick with your original cylinder, you'll win one time in three. So for every three coins you bet, you can expect to get two coins back - losing an average of 1/3 coin each go.

But if you switch you can expect to win two times out of three. But you've had to double your stake, while you're still only going to win one coin. But this is still better - over three games you'll bet six coins, but you'll also win back six coins (your two coins and the one coin from the showman, twice). So on average you break even - much better than losing 1/3 of a coin each go.


Quote:
Originally Posted by pdurrant View Post
Question 2: Should you agree to continue with this game after this accident?
It's hard to answer this question until you've answered Question 3 - I really should have put it last, not here. Sorry!

The answer is,
Spoiler:
it doesn't matter, although your strategy must change, see Q3.


Quote:
Originally Posted by pdurrant View Post
Question 3: If you do continue with this game, are you more likely to be better off moving your bet and doubling it, staying put, or does it not make any difference which you do?
Spoiler:
It seems at first sight that you're in the same situation as in Question 1 - that you should double your bet and switch to the other cylinder. Actually, if you do this, on average you'll lose 1/2 coin per game, while if you stick with your original cylinder, you'll break even.

What's the difference? After all, in both cases, a cylinder that you didn't choose has been shown to not have the coin under it.

The difference is that in this case, there was not a 100% chance that the cylinder would not reveal the coin.

Let's look at what could happen when a random cylinder is knocked over.

If it's the cylinder you bet against, the game is void. Either you or the showman would not want to continue in that circumstance, although if it showed the coin you might argue that you weren't going to switch....

But let's consider what happens when the random cylinder knocked over isn't the one next to your coin, since that's the case we're looking at. There are three cases to consider:
  1. If you have bet against the cylinder hiding the coin (1/3 chance), the chance that the cylinder knocked over isn't hiding the coin is 100% (1/1). Total chance of this happening: 1/3
  2. If you have bet against a cylinder that isn't hiding the coin (2/3 chance), the chance that the cylinder knocked over is hiding the coin is 50% (1/2). Total chance of this happening: 2/3 * 1/2 = 1/3
  3. If you have bet against a cylinder that isn't hiding the coin (2/3 chance), the chance that the cylinder knocked over also isn't hiding the coin is 50% (1/2). Total chance of this happening: 2/3 * 1/2 = 1/3

We know that case 2 hasn't happened. The two remaining possibilities are equally likely. So in this case, the chance that the other remaining cylinder is hiding the coin is 1/2, not 2/3.

Sticking with your cylinder will win one time in two. Swapping will win one time in two. But since you have to double your stake to swap, swapping will, oover two average games, cost four coins but only win three. Sticking where you are will cost two coins and win two coins.

So sticking breaks even, and swapping loses half a coin per game.

Another way of looking at this, to understand why the odds are different: If you've chosen the right cylinder to start with, the chance of the random other cylinder revealing no coin is much higher than if you have chosen one of the wrong cylinders. Since the random cylinder didn't reveal a coin, it's more likely than it was before that you have chosen the right cylinder to start with.
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