Question

4. If you place a $1 bet on a number of a roulette wheel, then either you win $35, with probability 1/38, or you lose $1, with probability 37/38. Let X denote your gain on a bet of this type.

(a) FindE[X]andSD(X).

Suppose you continually place bets of the preceding type. Show that

(b) The probability that you will be winning after 1000 bets is approxi- mately 0.39.

(c) Theprobabilitythatyouwillbewinningafter100,000betsisapprox- imately 0.002.

Answer 1

a)

x	P(x)	xP(x)	x2P(x)
35	1/38	0.921	32.237
-1	37/38	-0.974	0.974
	total	-0.053	33.211
	E(x) =μ=	ΣxP(x) =	-0.0526
	E(x2) =	Σx2P(x) =	33.2105
	Var(x)=σ2 =	E(x2)-(E(x))2=	33.208
	std deviation=	σ= √σ2 =	5.7626

from above :

E(X)=-0.0526

SD(X)=5.7626

b)

for 1000 bets; expected win=1000*-0.526=-52.63

and std deviation =5.7626*sqrt(1000)=182.23

hence from normal approximation ;

probability that you will be winning after 1000 bets =P(X>0)=P(Z>(0-(-52.63))/182.23)=P(Z>0.29)=~0.39

c)

for 100000 bets; expected win=100000*-0.526=-5263.16

and std deviation =5.7626*sqrt(1000)=1822.30

hence from normal approximation ;

probability that you will be winning after 100000 bets =P(X>0)=P(Z>(0-(-5263.16))/1822.3)=P(Z>2.89)=~0.002