Question

# A game is played. One dollar is bet and 3 dice are rolled. If the sum...

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.

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

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