Question

There are 6 closed boxes on the table. Two of them are non-empty, the rest 4 are empty. You open boxes one at a time until you find a non-empty one. Let X be the number of boxes you open.

(i) Find the probability mass function of X.

(ii) Find E(X) and V ar(X).

(iii) Suppose each non-empty box contains a $100 prize inside, but each empty box you open costs you $50. What is your expected gain or loss in this game?

Answer 1

i)

probability mass function of X is given below"

P(X=1)=P(1st is non empty)=2/6=1/3

P(X=2)=P(1st not but second is)=(4/6)*(2/5)=4/15

P(X=3)=(4/6)*(3/5)*(2/4)=1/5

P(X=4)=(4/6)*(3/5)*(2/4)*(2/3)=2/15

P(X=5)==(4/6)*(3/5)*(2/4)*(1/3)*(2/2)=1/15

ii)

x	f(x)	xP(x)	x2P(x)
1	1/3	0.333	0.333
2	4/15	0.533	1.067
3	1/5	0.600	1.800
4	2/15	0.533	2.133
5	1/15	0.333	1.667
	total	2.333	7.000
	E(x) =μ=	ΣxP(x) =	2.3333
	E(x2) =	Σx2P(x) =	7.0000
	Var(x)=σ2 =	E(x2)-(E(x))2=	1.5556

iii)

  expected gain or loss in this game

=(1/3)*100+(4/15)*(100-50)+(1/5)*(100-100)+(2/15)*(100-150)+(1/15)*(100-200)=33.33