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) | x^{2}P(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(x^{2}) = | Σx^{2}P(x) = | 8.3889 | |
Var(x)=σ^{2} = | E(x^{2})-(E(x))^{2}= | 8.015 | |
std deviation= | σ= √σ^{2} = | 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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