Suppose you draw 7 cards out of a deck of 52 What is the pro

Suppose you draw 7 cards out of a deck of 52. What is the probability that you will have (a) exactly 5 cards of one suit? (b) At least 5 cards of one suit?

Solution

a) one suit contains 13 cards

the number of ways a person can pick exactly 5 cards from 1 suit = ¹³C₅

since the other cards\' suit must not be of the same suit, they can be picked in ³⁹C₂

cards without any conditions can be picked in ⁵²C₇

the probability that the person will have exactly 5 cards of one suit =[¹³C₅X³⁹C₂]/⁵²C₇₌ = 0.007128

b) At least 5 cards of one suit = exactly 5 cards of one suit + exactly 6 cards of one suit + exactly 7 cards of one suit

for exactly 5 cards of one suit = [¹³C₅X³⁹C₂]/⁵²C₇

the number of ways a person can pick exactly 6 cards from 1 suit = ¹³C₆

since the other cards\' suit must not be of the same suit, they can be picked in³⁹C₁

for exactly 6 cards of one suit = [¹³C₆X³⁹C₁]/⁵²C₇

the number of ways a person can pick exactly 7 cards from 1 suit = ¹³C₇

for exactly 7 cards of one suit = [¹³C₇]/⁵²C₇

the probability that the person will have at least 5 cards of one suit =[¹³C₅X³⁹C₂]/⁵²C₇ + [¹³C₆X³⁹C₁]/⁵²C₇ +[¹³C₇]/⁵²C₇₌ 0.00764