Answer to Question #126149 in Statistics and Probability for sam

Question

Pull 5 cards from a standard deck, replacing and shuffling each time, and count the number of face cards.  Create probability distributions 

Answer 1

Solution:

There are 12 face cards in standart desk (52 cards).

Let p(t):=P(X=t) be the probability that there are t face card on the hand: t$\in${0,1,2,3,4,5}

p(0)=m/n,

m=binomial(40,5)=40!/5!/35!=658008,

n=binomial(52,5)=52!/5!/47!=2598960,

p(0)=658008/2598960=2109/8330,

p(1)=m/n,

m=binomial(12,1)*binomial(40,4)=12!/1!/11!*40!/4!/36!=1096680,

n=binomial(52,5)=2598960,

p(1)=1096680/2598960=703/1666,

p(2)=m/n,

m=binomial(12,2)*binomial(40,3)=12!/2!/10!*40!/3!/37!=652080,

n=binomial(52,5)=2598960,

p(2)=652080/2598960=209/833,

p(3)=m/n,

m=binomial(12,3)*binomial(40,2)=12!/3!/9!*40!/2!/38!=171600,

n=binomial(52,5)=2598960,

p(3)=171600/2598960=55/833,

p(4)=m/n,

m=binomial(12,4)*binomial(40,1)=12!/4!/8!*40!/1!/39!=19800,

n=binomial(52,5)=2598960,

p(4)=19800/2598960=165/21658,

p(5)=m/n,

m=binomial(12,5)=12!/5!/7!=792,

n=binomial(52,5)=2598960,

p(5)=792/2598960=33/108290,

Answer:

p(0)=2109/8330

p(1)=703/1666

p(2)=209/833,

p(3)=55/833,

p(4)=165/21658,

p(5)=33/108290.