Question

1. We draw randomly without replacement 3 balls from an urn that contains 3 red and 5 white balls. Denote by X the number of red balls drawn. Find the probability distribution of X, its expected value, and its standard deviation.

Answer 1

Probability distribution of X :

P(X=0)=P(0 red and 3 white) =(3C0)*(5C3)/(8C3) =1*10/56=5/28

P(X=1)==P(1 red and 2 white) =(3C1)*(5C2)/(8C3) =3*10/56=15/28

P(X=2)=P(2 red and 1 white) =(3C2)*(5C1)/(8C3) =3*5/56=15/56

P(X=3)=P(3 red and 0 white) =(3C3)*(5C0)/(8C3) =1*1/56=1/56

from above:

x	f(x)	xP(x)	x2P(x)
0	5/28	0.0000	0.0000
1	15/28	0.5357	0.5357
2	15/56	0.5357	1.0714
3	1/56	0.0536	0.1607
	total	1.1250	1.7679
	E(x) =μ=	ΣxP(x) =	1.1250
	E(x2) =	Σx2P(x) =	1.7679
	Var(x)=σ2 =	E(x2)-(E(x))2=	0.5022
	std deviation=	σ= √σ2 =	0.7087

expected value E(X) =1.1250

and standard deviation SD(x) =0.7087