Question

The deck for a card game contains 30 cards. 10 are red, 10 yellow, 5 blue, and 1 green, and 4 are wild cards. Each player is randomly dealt a five-card hand. a) What is the probability that a hand will contain exactly two wild cards? b) What is the probability that a hand will contain two wild cards, two red cards, and one blue cards?

Answer 1

Answer: a) 0.1095     b) 0.0095

Step-by-step explanation:

Given : The deck for a card game contains 30 cards.

10 are red, 10 yellow, 5 blue, and 1 green, and 4 are wild cards.

Each player is randomly dealt a five-card hand.

Number of ways to choose 5 cards out of 30 = $C(30,5)=\dfrac{30!}{5!25!}=142506$

a)  Cards other than wild card = 30-4=26

Number of ways to choose exactly two wild cards = $C(26,3)\timesC(4,2)$

$=\dfrac{26!}{3!23!}\times\dfrac{4!}{2!2!}\\\\=15600$

Probability that a hand will contain exactly two wild cards = $\dfrac{15600}{142506}=0.1095$

b) Number of ways to choose two wild cards, two red cards, and one blue cards = $C(4,2)\times C(10,2)\times C(5,1)$

$=\dfrac{4!}{2!2!}\times\dfrac{10!}{2!8!}\times5=1350$

Probability that a hand will contain two wild cards, two red cards, and one blue cards = $\dfrac{1350}{142506}=0.0095$