Life and luck

Question:

You and your friend are caught by gangsters and made to play a game to determine if you should live or die. The game is simple.

There is a deck of cards and you both have to choose a card. You can look at each other’s cards but not at the card you have chosen. You both will survive if both are correct in guessing the card they have chosen. Otherwise, both die.

What is the probability of you surviving if you and your friend play the game optimally?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Solution:

We know, A and B have picked a card at random from a deck. A can see B’s card and vice versa. So, A knows (s)he has not picked B’s card, but apart from that, (s)he knows that the card is equally probable to be any of the other 51 cards. So, if A guesses B’s card, they lose. But if A guesses any other card, there’s a 1/51 chance that A is right. This also implies that total probability of success <= 1/51.

A’s aim now is to tell any card apart from B’s card that gives B the most information about B’s own card. So they can plan beforehand as follows:

Consider the sequence of cards Clubs 1-13, Diamonds 1-13, Hearts 1-13, Spades 1-13. A will tell the card after B’s card in this sequence. (If A says 4 of Hearts, it means B has 3 of Hearts. If A says Ace of Clubs, it means B has King of Spades)

With A’s guess, which is always different from B’s card, B gets to know exactly which card (s)he has and can always guess correctly. So the probability of success is 1/51, which is the maximum achievable.

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