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Problem 329

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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.

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素数青蛙

苏珊有一只素数青蛙。
她的青蛙在500个标有1至500的方格上跳来跳去。它每次等概率地往左或往右跳一格，而且它不能跳出1到500这个范围。
（如果它跳到了两端，它将自动在下一步跳到唯一可行的那一格。）

当青蛙位于标有素数的方格时，它有2/3的概率会叫一声“P”（素数PRIME），有1/3的概率叫一声“N”（非素数NOT PRIME），叫完之后再跳下一步。
当青蛙位于标有非素数的方格时，它有1/3的概率会叫一声“P”，有2/3的概率叫一声“N”，叫完之后再跳下一步。

如果青蛙的初始位置等概率地位于任意方格，而且苏珊听到了青蛙前15次的叫声，那么苏珊听到的序列为PPPPNNPPPNPPNPN的概率是多少？

将你的答案表示为最简分数p/q。

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