| yes, you have a better chance.
to start out, each envelope has a 1/3 chance of having the bill. So when you get the first envelope , you have a 1/3 chance of having the bill
There is a 2/3 chance that the bill is in the other person's possession. After discarding of one envelope, the chance of that envelope having the bill is zero. the odds must still add up to one, so there is still a 2/3 chance of the other envelope having the bill. So, by switching envelopes, you have a better chance of having the bill
Another way to look at it is like this: By switching, you are in essence taking the two other envelopes. you already knew that one of them didnt have a bill, so the fact that one of them is discarded is inconsequential. there is a 2/3 chance that the bill is in one of those two envelopes, and an empty envelope will be discarded regardless |