Recently, I got stuck with the following task
The player twists 50 times a roulette wheel (a circle with 5 sectors of equal size, on these sectors there are numbers from 1 to 5) and each time equally likely he gets a number from one of these sectors. This is his profit. Find the probability that he'll win at least $170.
I'm accustomed that problems of the theory of probability are about patterns recognition, and each problem we should reduce to the problem that we already know how to solve, to some known "story" or distribution. But that task doesn't look like smth I've ever came across. And I didn't manage to think up, how to solve it.
Could you please give me any hints for solving, or references to similar problems' solution (if that problem is a known one). Thanks a lot in advance!
Let $X_1, \ldots X_{50}$ denote the value of the individual rolls. You are then looking at the distribution of $S = \sum_{k=1}^{50} X_k$.
If you define $Y_i = aX_i+b$ and find $a,b \in \mathbb{R}$ such that $Y_i$ has 0 mean and unit variance, you would be able to approximate $S/{50}$ using a standard normal distribution using the Central Limit Theorem.