Find probability that mean score of dice thrown 70 times is less then 3.3

Question

An unbiased dice is thrown once. Mean = 7/2, Variance = 35/12

The same dice is thrown 70 times

1) Find the probability that the mean score is less than 3.3

2) Find the probability that the total score exceeds 260

Answer 1

For (2), I suppose you are to use a normal approximation to the sum $T = \sum_{i=1}^{70} X_i,$ where $X_i$ is the number of spots showing at the $i$th roll.

You have $\mu_T = E(T) = 70(7/2).$ Also, because rolls are independent $Var(T) = 70(35/12)$ and from that you can find $\sigma_T = SD(T).$

Then $$P(T > 260) = P\left(\frac{T - \mu_T}{\sigma_T} > \frac{260 - \mu_T}{\sigma_T} \right) \approx 1- P\left(Z \le \frac{260 - \mu_T}{\sigma_T} \right),$$ where $Z$ is standard normal.

(1) Similar, except that $E(\bar X) = 7/2$ and $SD(\bar X) = \sqrt{(35/12)/70}.$