Prime Frog
Susan has a prime frog.
Her frog is jumping around over $500$ squares numbered $1$ to $500$. He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range $[1;500]$.
(if it lands at either end, it automatically jumps to the only available square on the next move.)
When he is on a square with a prime number on it, he croaks 'P' (PRIME) with probability $2/3$ or 'N' (NOT PRIME) with probability $1/3$ just before jumping to the next square.
When he is on a square with a number on it that is not a prime he croaks 'P' with probability $1/3$ or 'N' with probability $2/3$ just before jumping to the next square.
Given that the frog's starting position is random with the same probability for every square, and given that she listens to his first $15$ croaks, what is the probability that she hears the sequence PPPPNNPPPNPPNPN?
Give your answer as a fraction $p/q$ in reduced form.