"There is no 3rd ash tray because the bar tender took it away. What are the odds? 2 in 3, right? WRONG!!! They are 50:50."

That's what it might seem, if there was no history.

If the barman tells you what has happened, and which one was originally chosen, the odds are 2 in 3 of a win if you change the original choice.

Look at it this way, if you choose one on the left, and move the others to the right, the chances of the prize being on your right are 2 in 3, and on your left 1 in 3. OK?

So if the barman removes one on the right, it doesn't alter the odds of the one you chose on the left, which is still 1 in 3.

There is a 2 in 3 chance of the prize being on the right still, and the barman has kindly removed one of the options, so there is only one choice for the 2 in 3 option on the right.

Look at the table I posted and it shows the results for the various choice and prize positions and you can see how it works.