# Math Help - Central Limit Theorem Application

1. ## Central Limit Theorem Application

A die is continuously rolled 65 -1 times. What is the probability that the total sum of all rolls does not exceed 225?

I know that the expected value for each roll is 7/2 so this means that the mean sum of all the rolls would be 64*(7/2) = 224.

Also, the variance for each roll is 35/12. How do I use this to find the std deviation in all the rolls?

2. ## Re: Central Limit Theorem Application

The variance of the sum is $n\sigma^2$ where $\sigma^2$ is the variance of 1 roll.

However to apply the central limit theorum you should be thinking about the properties of the sample mean. if the total does not exceed 225, what can you say about the sample mean?

Spoiler:

$P(total< 225) = P(sample~mean < \frac{225}{64})$

by CLT, sample mean approximately follows $N(3.5,\frac{\sigma^2}{n})$