Question

We draw 6 cards from a 52 card deck and let X = the number of heart cards drawn.

a. What is the expected value of X?
b. What is the variance of X? What is the standard deviation of X?

Answer 1

let X = the number of heart cards drawn.X can vary from 0 to 6.

there are 13 hearts cards, and 39 non hearts cards

P(X=0) = (13C0)*39C6 / 52C6

P(X=1) = 13C1 * 39C5 / 52C6

P(X=2)=13C2 * 39C4 / 52C6

P(X=3)=13C3 * 39C3 / 52C6

P(X=4)=13C4 * 39C2 / 52C6

P(X=5)=13C5 * 39C1 / 52C6

P(X=6)=13C6 * 39C0 / 52C6

X	P(X)	X*P(X)	X² * P(X)
0	0.1603	0	0
1	0.3677	0.367652	0.367652
2	0.3151	0.63026	1.26052
3	0.1284	0.385159	1.155476
4	0.0260	0.104097	0.416388
5	0.0025	0.012327	0.061636
6	0.0001	0.000506	0.003034

	P(X)	X*P(X)	X² * P(X)
total sum =	1	1.5	3.264706

a)

expected value= E[X] = Σx*P(X) =            1.5
          
b)

E [ X² ] = ΣX² * P(X) =            3.264705882
          
variance = E[ X² ] - (E[ X ])² =            1.0147
          
std dev = √(variance) =            1.0073