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?

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)