You leave your apartment groggily one morning, closing the door behind you. Suddenly, you are hit by a terrifying question: Do you have your keys, or are you now locked out?
You stand there thinking about it for a few seconds, before deciding that yes, you probably have your keys, further estimating that 80% of the time, you have them. You also decide that there is an equal chance of your keys either being in your left pocket or your right pocket, and if they aren’t in either pocket then you don’t have them at all.
Slowly, perversely enjoying the sweat, you slide your hand into your right pocket, and find that your keys are not there. What should you now think is the probability that your keys are in your left pocket?
Two-thirds. To solve this without (much) explicit calculation, imagine all the scenarios where your right pocket is empty. 60% of the time, your right pocket will be empty (20% of the time because you forgot the keys, and 40% of the time where you remembered the keys but they were in the left pocket). The given information puts you in this 60%. Of this 60%, 2/3 of the time you actually have the keys.