GeorgeSanJose —"I can't possibly remember my probability classes and definitely not Bayes Theorem. I just listed out all the possible combinations for the three cards. ... Does that sound correct?"I don't know ... you lost me somewhere in the middle of your post.
The cowboy has two strategies: When asked to make a choice after the gambler flips a red/red or a green/green card, he can either stand pat with his first decision, or switch and point to another card.
If he stands pat, his odds of winning are 1/3, the same as it was before the gambler flipped the card.
If he always switches his selection, his odds of initially picking a red/red or a green/green card on the first selection are 2/3. But now he wins because the gambler must flip the remaining red/red or green/green card and the cowboy wins after switching has selection to the remaining card, which must be the card with differing face colors.
So the cowboys strategies and odds of winning are —
- Stand pat and win one out of three; or
- Always switch picks and win two out of three.
This could be solved by a modification of Bayes’ theorem using three variables, but I don’t think it’s worth the effort.