https://en.wikipedia.org/wiki/Monty_Hall_problem
We have a card game with 3 cards. 2 blue cards (B) and 1 red card (R).
The game has two players. A host and a contestent. The host lays out the 3 cards infront of the contenstent, knowing which one is R. The goal is for the contestent to find R. The game two rounds.
- In the first round the contestent picks a card among the 3 which they think is R.
- In the second round the host will point another card that isn't R and disclose this to the contestent. The contestent now has the option of re-picking a card which they think is R amont the 2 remaining ones.
Here is an example playthrough
The host lays out the cards in the following order (unknown to the contestent ofcourse)
1 2 3
R B B
The contestent picks card number 2 (by random choice). The host then points to 3 and discloses that is B. The contestent now has the option of changing their pick to 1 or stay on 2. They choose to switch to 1 and win the game since it was R.
The best strategy in the game is to always change your pick when giving the choice in round 2. This gives you a 2/3 chance of winning the game (!) vs 1/3 if not. Let us look at why.
In the 1st round the contestent looks at 3 random cards.
1 2 3
If they pick 1, there is a 1/3 probability that, that is R. This corresponds to a 2/3 probability that either 2 or 3 is R. as well.
In the 2nd round the host will disclose that either 2 or 3 is B. Since we know from before that there is a 2/3 probability that 2 or 3 is R and now we know which of 2 or 3 is definitely B (the host just told us) this means that there is a 2/3 probability that the card the host did not pick is R.
If we change our pick we therefore up our chances from 1/3 to 2/3.