Product of dice rolls (expected value of the number of rolls)

We can solve this problem with the idea of using recursion with states. We introduce three states:

$S_1 =$ when the product is 1 (i.e. only one divisor - this is where we start at the beginning),

$S_p$ = when the product is a prime number (i.e. exactly two divisors),

and

$S_f =$ the end of rollig, so when the product has at least three divisors.

Furthermore, let:

$E_1 = E(X|S_1) =$ the expected value of rolls if we start from state $S_1$

and

$E_p = E(X|S_p) =$ the expected value of rolls if we start from state $S_p$ .

We can write the following equations:

$E_1 = \tfrac{1}{6}\cdot (1+E_1) + \tfrac{1}{2}\cdot (1+E_p) + \tfrac{1}{3}\cdot 1 = 1 + \tfrac{1}{6}\cdot E_1 + \tfrac{1}{2}\cdot E_p$

and

$E_p = \tfrac{1}{6}\cdot (1+E_p) + \tfrac{5}{6}\cdot 1 = 1 + \tfrac{1}{6} E_p$ ,

which we can solve easily. The solution is $E_1 = \frac{48}{25} = 1.92$ .