2 Card Game

Consider this game: I write two distinct numbers on two separate cards, and allow you to pick one. You can look at the number written on the card, and you can decide if you want to switch cards. Your objective is to get the card with the higher value written on it. Is there a strategy that lets you win with >50% success?

Surprisingly, the answer is yes! Consider the following strategy: We sample randomly from any distribution , say the standard normal distribution. We compare this value with the value on the card, and change cards whenever our sampled value is higher than the value on the card.

So what’s the chance of winning with this strategy? If we let the two numbers on the two cards be and , with , we can consider a few cases for where our sampled ends up. If , then no matter which card we picked, we would switch, and the chance of success is 50%. Likewise, if , then we would always keep the card we picked, so the chance of success is still 50%.

If however, the value we picked happened to be between the two values and , then we would always switch to the larger card, so the chance of success is 100%. This gives us a total winning chance of .