# Normal distribution for approximation of expected profit

Gambling: Crown and Anchor. Crown and Anchor is a game that is sometimes played at charity casinos or just for fun. It can be played with a “wheel of fortune" or with 3 dice, in which each die has its 6 sides labelled with a crown, an anchor, and the four card suits club, diamond, heart and spade, respectively. You bet an amount (let’s say \$1) on one of the 6 symbols: let’s suppose you bet on “heart". The 3 dice are then rolled simultaneously and you win \$t if t hearts turn up ($t = 0, 1, 2, 3$).

Let $X$ represent your profits from playing the game n times. Give a normal approximation for the distribution of $X$.

How should I do this problem?

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You have a discrete probability distribution for one game. There is a certain probability of being $-1,0,+1,$ and $+2$. Calculate those probabilities, then the expected value and variance. Your normal distribution (for $n$ large) is that the mean is $n$ times the mean of the distribution and the variance is $n$ times the variance of the distribution.