A game of chance?
You and your friend are playing a game, where you each randomly select one of 4 cards, each labelled with one different number between 1-4, not showing each other what you received. Before playing, you both agree that the winner will win a substantial prize, a big enough reward/consequence that you both really want to win. The player with the higher card wins. You draw the 2, and your friend draws either the 1,3 or 4. Your friend then offers you a choice, if you want to switch cards with each other, you can.
Should you switch cards?
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1 solution
At first, it seems that switching would be a smart idea since there is a 67% chance that your friend has either the 3 or 4. However, you need to view your friend's standpoint to understand what to do. Here are the different possibilities of what your friend has.
A) They have the 4, and therefore would not offer the switch, because then they will 100% lose. B) They have the 3, and since they know that you have the 1,2 or 4, they will keep the 3, because they then have a 2/3 chance of winning, and therefore would not offer the switch. C) They have the 2, which is impossible since you have the 2. D) They have the 1, and will then 100% offer the switch because if they don't they will 100% lose.
Using this information, your friend will only offer the switch if they have the 1, and therefore you should not switch.