Probability

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An ordinary deck of playing cards has 4 suits, with 13 cards of each suit. In many games, each of 4 players is dealt 13 cards at random.
a. What is the probability that such a hand has each of the following?
1. Exactly 5 spades

2. Which is more probable, getting all 4 aces or getting all 13 cards of the same suit? Justify your answer.

  • Probability -

    let S be the event of being dealt a spade,
    x the event of a non-spade

    One such 5 spade hand could be
    SSSSSXXXXXXXX
    The prob of that is
    (13/52*12/51*11/50*10/49*9/48)(39/47*38/46*37/45*36/44*35/43*34/33*33/32*32/31)

    a quicker way would be to say
    prob(exactly 5 spades)
    = ( C(13,5)*C(39,8)/C(52,13)
    = .12469

    prob (4 aces)
    = 4C(4,4)*C(48,9)/C(52,13) = appr .01056
    prob(13 of same suit)
    = 4/C(52,3) = very small
    not even close!

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