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16 November, 11:53

Twelve people each have a standard deck of cards. Each person shuffles the deck and draws two cards (without replacement). Anyone who draws at least one diamond wins a prize. To the nearest whole number, about how many of the twelve will win a prize? What is the probability that at least one person will win a prize? (Answer correct to four decimal places.) What is the probability that two people will win a prize?

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  1. 16 November, 12:57
    0
    5

    0.9990

    0.0401

    Step-by-step explanation:

    1) Probability (diamond) = 13/52 = 1/4

    Probability (not diamond) = 1 - 1/4 = 3/4

    Probability of atleast one diamond = 1 - probability of no diamonds

    = 1 - (¾ * ¾) = 1 - 9/16 = 7/16

    Expected no. = np = 12 * 7/16

    = 5.25 = 5 (nearest whole no.)

    2) let X be the no. of people winning the prize.

    Then X ~ B (12, 7/16)

    [ X follows a Binomial Distribution with 12 trials and probability of success (winning prize) = 7/16 ]

    P (atleast 1 winning) = 1 - P (X=0)

    = 1 - (9/16) ^12

    = 0.9989966087

    = 0.9990 (4 dp)

    3) P (X=2) = 12C2 * (7/16) ² * (9/16) ^10

    = 0.0400613268

    = 0.0401 (4 dp)
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