Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.
[Math] Statistics question on probability
diceprobabilitystatistics
diceprobabilitystatistics
Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.
Best Answer
Let $Y$ be the sum of the $100$ dice. This is $X_1+X_2+\cdots +X_{100}$, where the $X_i$ are the individual results.
We need to compute the mean and the variance of the $X_i$. Each $X_i$ has mean $3.5$. For the variance of an $X_i$, use the fact that the variance is $E(X_i^2)-(E(X_i))^2$. To compute $E(X_i^2)$, find $\frac{1}{6}(1^2+\cdots+6^2)$. After a while, I think we get that the variance of $X_i$ is $\frac{35}{12}$.
So the mean of $Y$ is $350$ and the variance, by independence, is $\frac{3500}{12}$.
Now we want the probability that the sum is $371$ or more. This is $1$ minus the probability that the sum is $370$ or less.
The $X_i$ are "nice" independent identically distributed random variables. It is reasonable to assume that the sum $X_1+\cdots+X_{100}$ has distribution that is well-approximated by a normal.
The probability that the sum is $\le 370$ is well-approximated by the probability that the approximating normal is $\le 370.5$. I assume you can do the rest. If you have trouble, please send a message.