This problem throws many people, even a number of ‘expert’
mathematicians. Suppose you first choose envelope A:

Round-the-world tickets are in

Host opens

If you stick

If you switch

Envelope A

B or C

Star prize

Booby prize

Envelope B

C

Booby prize

Star prize

Envelope C

B

Booby prize

Star prize

The star prize is equally likely to be in any of the three
envelopes. You therefore have a 2 in 3 chance of winning the
round-the-world tickets if you change your mind, compared to a 1 in 3 chance
if you stick. Consequently, you are better off opening the envelope you
didn’t choose first.