The probability that the product of 100 die rolls $\le$ $a^{100}$ for $1 < a < 6$

Question

Suppose that a fair $6$ sided die is rolled 100 times. Let $X_i$ be the value obtained on the $i$th roll. Compute an approximation for $ P\left(\prod X_i \le a^{100}\right)$ for $1 < a < 6$.

I'm thinking about using Central Limit Theorem but not sure how. Any help is appreciated!

Answer 1

Hints:

• To use the central limit theorem, you need to translate the product into a sum. There is a natural way of doing this

• You will need to calculate the mean and variance or standard deviation for these new random variables

• You can then use the normal distribution (or, if you prefer, the log-normal distribution) to approximate your inequality