[HarryT]

Originally Posted by pdurrant

You decide to have a go. You place your bet against the right-hand cylinder. The showman lifts the centre cylinder to reveal nothing. You now have to choose whether to double your stake and swap to the left hand cylinder, or leave your bet where it is.

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 and, 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.