Monty Hall, counted by hand
Three doors, one car. You pick door 1. The host, who knows where the car is, opens another door with a goat behind it and offers you a switch. Should you switch?
Arguments about this go on forever, so here is the slow way: list every case.
| Car behind | Host opens | Stay wins? | Switch wins? |
|---|---|---|---|
| Door 1 | 2 or 3 | yes | no |
| Door 2 | 3 | no | yes |
| Door 3 | 2 | no | yes |
Each row is equally likely, because the car was placed before you chose. Staying wins in one row out of three, switching wins in two.
The detail that matters
The answer depends on the host always opening a goat door. If he opened a door at random and it merely happened to show a goat, the odds would be 50/50. The host's knowledge is what moves probability onto the remaining door.