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

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)=(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)*C(4,2)`

`=(26!)/(3!23!)*(4!)/(2!2!)\\\\=15600`

Probability that a hand will contain exactly two wild cards =
`(15600)/(142506)=0.1095`

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

`=(4!)/(2!2!)*(10!)/(2!8!)*5=1350`

Probability that a hand will contain two wild cards, two red cards, and one blue cards =
`(1350)/(142506)=0.0095`