Question

Ten people each have a deck of 52 playing cards. Each person randomly selects 15 cards from their deck without replacement. What is the probability that exactly three of the ten people select exactly five clubs?

(A) 0.1449 (B) 0.1508 (C) 0.1653 (D) 0.1781 (E) 0.1826

Answer 1

Solution: Option( D ) = 0.1781

total number of cards = 52

number of cards to be selected by a person = 15

number of clubs in a pack = 13

total number of ways of selecting 15 cards = ⁵²C₁₃ = 52! / 13!(52-13)! = 4481381406320

number of ways of selecting exactly 5 clubs = ¹³C₅ * ³⁹C₁₀ = 1287 * 635745396 = 818204324652

probability of ways of selecting exactly 5 clubs = 818204324652 /  4481381406320 = 0.1825

now,

number of people to be selected = n = 10

probability of a random person selecting exactly 5 clubs = p = 0.1825

binomial probability distribution

Formula:

P_{(k out of n )}= n!*p^k * q^n-k / k! *(n - k)!

P( x = 3 ) = 10!*0.1825³ * 0.8175^10-3 / 3! *(10 - 3)!

= 0.1781