Decks of Pinochle cards have a total of 48 cards and consist

Decks of Pinochle cards have a total of 48 cards and consist of 6 cards each of nines, tens, jacks, queens, kings, and aces with there being two of each suit of each denomination (for example, there are 2 aces each of diamonds, clubs, hearts, and spades for the total of 8 aces). Suppose that you are dealt a 12-card hand from a deck of Pinochle cards. What is the probability that:

a) you are dealt exactly 4 hearts?

b) you are dealt at most 2 aces?

Solution

There are in total 48 cards.

6 cards each of 9, 10, J, Q, K and A.

Of those Ace willbe 2 in dice hearts, etc.

Dealt with 12 cards.

a) There are totally 12 hearts

Prob for a heart = P(Heart) = p = 12/48 =0.25

q = P(non heart) = 0.75

There are only 2 outcomes heart and non heart.

X - No of hearts

Hence Prob that dealth with 4 hearts =

P(X=4 in 12 cards)

= 12C4(0.25)⁴(0.75)⁸

= 0.1936

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b) There are totally 8 aces in 48 cards.

P(atmost 2 aces)

= P(0 ace)+P(1 ace)+P(2 aces)

Prob for each ace = 8/48 = 0.1667

If y is the no of aces Y is binomial with p = 0.1667 and n = 12

Reqd probability

= P(0)+P(1)+P(2)

= 0.1121+0.2691+0.2961

= 0.6773