Question

A game is played. One dollar is bet and 3 dice are rolled. If the sum of the eyes of the dice is smaller than 5, one wins 20 dollar otherwise one looses the 1 dollar.

Find the approximation of the distribution of the total wins after 200 games. What is the Probabilty that the total win is positive. Use the CLT and state why.

Answer 1

P( eyes of the dice is smaller than 5)=4/216=1/54

hence

x	P(x)	xP(x)	x2P(x)
20	1/54	0.370	7.407
-1	53/54	-0.981	0.981
	total	-0.611	8.389
	E(x) =μ=	ΣxP(x) =	-0.6111
	E(x2) =	Σx2P(x) =	8.3889
	Var(x)=σ2 =	E(x2)-(E(x))2=	8.015
	std deviation=	σ= √σ2 =	2.8312

hence expected win after 200 games = 200*(-0.6111)=-122.22

and standard deviation =2.8312*sqrt(200)=40.0386

as sample size is greater then 30 ; therefore we can use normal approximation

therfore from normal approximation: P( that the total win is positive )=P(X>0)=P(Z>(0-(-122.22))/40.0386)=P(Z>3.05)

=0.0011