# What is the probability that the second card drawn is a spade if the first card drawn is an ace?

1. 1st card drawn is not replaced.

2. 52 cards in a deck.

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• Anonymous
8 years ago

(1/4) of the time the first card will be the spade Ace and the probability of the next card being a spade is (12/51)

(3/4) of the time the first card wqill be a different ace and the probability of the second card drawn being a spade is (13/51)

((1/4) * (12/51)) + ((3/4) * (13/51)) = (12 + 39) / 204) = (51 / 204) = (1/4) <----- Answer

This is intuitively obvious since (1/4) of aces are spades and (1/4) of non-aces are spades.

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• Hmm, are the other posters reading too much into this question?

If the first card is an ace, it could be the ace of spades (1/4 chance) or one of the other aces (3/4 chance). The probability that the next card is a spade is 1/4 - pulling an ace first has no effect on the result.

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OK, I see Gerry has made a correction. :)

• Consider the cases: if your ace is a spade, then you have 51 cards remaining, but only 12 of which are spades. Hence P(Spade) = 12/51

If your ace is not a spade, then you have 51 cards remaining and 13 spades remaining, hence P