Question

In: Statistics and Probability

You draw two cards from a standard deck without replacement and without looking at them. Then...

You draw two cards from a standard deck without replacement and without looking at them. Then the third card is drawn and it is revealed: it is the ace of hearts. What is the conditional probability that your two cards, formerly hidden, are aces?

Expert Solution

There are 4 aces and 48 other cards in a deck.

There are three cases possible :

First two card are aces or one of first two cards is ace or none of first two card is ace.

Case: First two card are aces

The probabilty of getting third ace and first two aces is

P(first two aces and third ace) =(4/52) * (3/51) * (2/50) = 24 /(52*51*50)

Case: One of first two cards is ace

The probabilty of getting third ace is

P(one of first two is an ace and third ace) =2 * (4/52) * (48/51) * (3/50) = 1152 / (52*51*50)

Case: None of first two cards is ace

The probabilty of getting third ace is

P(none of first two is an ace and third ace) = (48/52) * (47/51) * (3/50) = 6768 / (52*51*50)

The probability that third card is an ace is:

P(third ace) = [24 /(52*51*50)]+ [1152 / (52*51*50)] + [9024 / (52*51*50)] = 10200 / (52*51*50)

The conditional probability that your two cards, formerly hidden, are aces is

P(first two cards ace |third ace) = P(first two aces and third ace) / P(third ace) = 24 / 10200 = 0.0024

orchestra answered 2 years ago

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