Question

There's a pack of cards with 13 cards that goes from A to K, A being the lowest of value and K being the highest. There are 2 players in the game Player 1 retrieved 6 cards from the pack (which he/she is allowed to look) and the rest of the cards goes in as a faced-down pile. Player 2 must draw 1 card at a time (without getting to look), and he stop if he thinks the card he held is the highest in the pile. If it's not the highest when he stop, Player 1 wins and vice versa.

a) How do Player 2 decide when to stop? How do he knows the next card he pick is the highest or not?

b) Post game, if you decide to play the game, will you be the one piling or the one choosing?

Answer 1

Let's call the following event X :A will definetly win if one of the six cards that he retrieved at the start is K. So probability of that happening is: Selecting K card, and selecting other 5 cards from the rest of the deck (12 cards:excluding K) i.e P(X) =  . = 0.46

Event Z = Probability of Player 2 winning: 'Event X does not happen' AND 'He stops at the right card'

Event M = Probability that Player 2 stops at the right card.

Card K could be anywhere in the pile of 7 cards. Event M is essentially choosing the right card (i.e. K) out of 7 cards.

So P(M) = $\frac{1}{7}$

P(Z) = P(X^c ${\bigcap }^{}$ M) = P(X^c) . P(M) = (1-0.46) x (0.143) = 0.077.

Since "Player 1 winning" and "Player 2 winning" are mutually exclusive and exhaustive events. Their respective probabilities sum to 1.

Event Y = Player 2 wins.

P(Y) = 1 - P(Z) = 0.923.

P(Player 1 wins) = P(Z) = 0.923

P(Player 2 wins) = P(Y) = 0.077

a.) Player 2 cannot possibly decide when to stop. For all he knows, the card K could not even be among the 7 cards faced-down pile. He cannot know if the next card that he picks is the highest or not. It's a probabilistic event in which he has to pick a correct card out of 7.

b.) Since,

P(Player 1 wins) = P(Z) = 0.923

P(Player 2 wins) = P(Y) = 0.077

I'll choose to pile because in that case probability of my winning will be very high.