A player rolls 3 dice (6 sides) simultanuously. If at least one 5 or 6 appeared he wins 2 dollars. In any other case he loses 6 dollars.
a) Find pdf of random variable $Y$ profit of player at a game
b) which is the expected profit of the player and variance?
c) If he played 20 games in a row he expects to win or lose and what ammount?
d) what is the minimum ammount that he must have with him every time he plays and wins so that he has benefit?
e) If the profit of the player comes from $Y_1=2Y$ or $Y_2=3Y+2$. Find mean and variance. And which of the $Y_1$ or $Y_2$ he has to accept?
My solution:
a) Let $X$ the number of 5 or 6 that appears.
$X~Binomial(3,1/3)$,
$P(X=0)=64/216$, $P(X=1)=96/216$, $P(X=2)=48/216$, $P(X=3)=8/216$.
Let $Y$=the profit of the plaayer an a game. $S_Y={-6,2}$,
$P(Y=-6)=64/216$, $P(Y=2)=(96+48+8)/216=152/216$
b) $E[Y]=-80/216$, $Var(Y)=13,344$
c) Let $Z$ the number of wins if he plays 20 games (random variable). $Z~Binomial(20,152/216)$,
$E(Z)=14,074$
d) For (d) is 0,37 dollars?
e) $E[Y_1]=2E[Y]=-0,74$, $E[Y_2]=3E[Y]+2=0,88$, $Var(Y_1)=4Var(Y)$, $Var(Y_2)=9Var(Y)$. Is $Y_2$ better than $Y_1$?