I recently read "The Curious Incident of the Dog in the Night-Time" by Mark Haddon. It is the story, written in the first person, about an autistic boy called Christopher who investigates the murder of his neighbour's dog.
One of the most enjoyable parts of the book for me was the sprinkling of mathematical puzzles which include the "Monty Hall" probability problem by Marilyn vos Savant. The problem goes like this:
You are on a game show and you are given the choice of three doors.
Behind one door is a car (the main prize).
Behind each of the other two doors is a goat (you lose).
You pick a door and the host (who knows what's behind the doors) opens another door which has a goat behind it.
The host asks you if you want to stick to your choice or to switch to the other unopened door.
Is it better to stick or switch doors?
If you think that the probability of winning the car is the same if you stick with your original choice or switch your choice to the other unopened door (0.5) then you are wrong!
The table below shows that the probability of winning by staying with your original choice is 1/3 and your probability of winning by switching to the other unopened door is 2/3.
You pick door 1, the host opens one of the losing doors.
+------+--------+--------+--------+---------------------+
| Game | Door 1 | Door 2 | Door 3 | Result              |
+------+--------+--------+--------+---------------------+
| 1    | Car    | Goat   | Goat   | Switch and you lose |
| 2    | Goat   | Goat   | Car    | Switch and you win  |
| 3    | Goat   | Car    | Goat   | Switch and you win  |
| 4    | Car    | Goat   | Goat   | Stick and you win   |
| 5    | Goat   | Goat   | Car    | Stick and you lose  |
| 6    | Goat   | Car    | Goat   | Stick and you lose  |
+------+--------+--------+--------+---------------------+
Staying with your original choice wins 1/3 of the time, while switching doors wins 2/3 times.