[pdurrant]

Originally Posted by HarryT

Question: When the showman lifts the initial cylinder, will he always choose to lift one that he knows the coin is not under? On the assumption that the answer to this is "Yes", then I'm going to tackle question 1 as follows:

Spoiler:

Let us call the three cylinders A, B, and C. You guess "A" and have (obviously) a 1/3 probability of being right. The chance of the coin being under "B" or "C" is 2/3. Now, the showman lifts "C" to reveal nothing. This leaves us still with a 1/3 chance of A being the correct guess, but now a 2/3 chance of B being correct. So, all being equal, you should select "B".

However, you will have to double your stake to do so. Double the stake for double the chance. If you stay where you are, you have a 1/3 chance of winning £1, and a 2/3 chance of losing £1. If you change your bet, you have a 2/3 chance of winning £1, but a 1/3 chance of losing £2.

So, stay where you are in, over the long term, the result will be:

+1/3*1 - 2/3*1 = -1/3

Change your bet: +2/3*1 - 1/3*2 = 0

So I reckon you should always change your bet. The long-term results of doing so will be that you break even.

Now, for question 2:

Spoiler:

The fact that now the cylinder has been lifted at random to reveal nothing changes the odds to 50:50 between the coin being under "A" or "B".

So stick with "A" - a 50:50 chance of winning £1 or losing £1. Change to "B" and the odds become a 50:50 chance of winning £1, or losing £2. Clearly you should leave your bet where it is, and again in the long term you should break even.

I've answered Q3 in part 1.

I've added some text to the question to make it explicit that the showman lifts a cylinder that he knows isn't covering the coin.

Both the answers you've given are right. I can't see your explicit answer to Q2, but given that you've got Q1 and Q3 right, the answer to Q2 is easy.