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10 September, 00:06

We are given a random collection of 3 cards. (a) What is the conditional probability that all 3 cards are spades, given that the collection contains the ace of spades? Find an approximate decimal. (b) What is the conditional probability that all 3 cards are spades, given that at least one card is a spade? Find an approximate decimal.

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  1. 10 September, 02:17
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    a) Assuming a standard deck of 52 cards. By definition of conditional probability:

    P (3 spades | ace of spades) = P (3 spades AND ace of spades) / P (ace of spades)

    We draw 3 cards, so there are (52 choose 3) = 22,100 total possible hands.

    There are (1 choose 1) * (51 choose 2) = 1275 possible hands with the ace of spades.

    There are (13 choose 3) = 286 possible hands containing 3 spades, and (1 choose 1) * (12 choose 2) = 66 of these contain the ace of spades.

    So we have

    P (3 spades AND ace of spades) = 66/22,100

    P (ace of spades = 1275/22,100

    and so

    P (3 spades | ace of spades) = 66/1275 = 0.0518

    b) By definition:

    P (3 spades | at least 1 spade) = P (3 spades AND at least 1 spade) / P (at least 1 spade)

    If you draw 3 spades, you already meet the requirement of having at least 1 spade, so

    P (3 spades AND at least 1 spade) = P (3 spades)

    and we already know there are 286 possible such hands.

    The event of getting at least 1 spades is complementary to the event of not getting any spades, meaning

    P (at least 1 spade) = 1 - P (no spades)

    There are 39 non-spade cards, and so (39 choose 3) = 9139 possible hands not containing spades.

    So we find

    P (3 spades) = 286/22,100

    P (at least 1 spade) = 1 - 9139/22,100 = 12,961/22,100

    which means

    P (3 spades | at least 1 spade) = 286/12,961 = 0.0221
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