A standard deck of 52 playing cards includes 4 each of Aces, Kings, Queens, Jack

Question

A standard deck of 52 playing cards includes 4 each of Aces, Kings, Queens, Jacks, and

tens through twos. Each of the four cards comes in one of four suits: spades, hearts,

diamonds, and clubs. Assume you are playing a hand of 328 poker, which has slightly

different rules than regular poker. The key difference is that in 328 poker, you will be

dealt six cards instead of five and all of the 3’s are removed from the deck. You may give

your answers as fractions, and there is no need to reduce them.

a. What is the probability of getting at least one picture card (King, Queen, or Jack)

in the six cards you are dealt?

b. What is the probability of getting four Aces in the six cards you are dealt?

c. What is the probability of being dealt a “flush” (all six cards are the same suit)?

d. Define D as a random variable that denotes the number of diamond-suited cards

in a player’s hand. Does D follow a binomial distribution? Why or why not?

Explanation / Answer

A.a) cards dealt=6

total picture cards=12

total deck size=48

to find probability of atleast one picture card.. P(X1)

P(X1)=1- P(X=0)

=1-36C6/48C6

=1-0.1587

=0.84

probability of atleast one picture card is 0.84 or 84%

A.b) probability of getting four Aces

P(X)=4C4*44C2/48C6

probability =0.000077

A.c) we have 4 suits

hence 4C1=4 choices

to have flush we need all cards of same suit

i.e, 12C6 =924 ways

probability of having flush

P(X)=4*924/48C6

probability =0.0003

A.d) we can say that D follows binomial distribution because having or not having diamonds in player hand are both mutually exclusive events which can be termed as p and q in binomial distribution

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