Question 51
Consider a number line, with a lily pad placed at each integer point. A frog is standing at the lily pad at the point $0$ on the number line, and wants to reach the lily pad at the point $2014$ on the number line. If the frog stands at the point $n$ on the number line, it can jump directly to either point $n+2$ or point $n+3$ on the number line. Each of the lily pads at the points $1, \cdots, 2013$ on the number line has, independently and with probability $1 / 2$, a snake. Let $p$ be the probability that the frog can make some sequence of jumps to reach the lily pad at the point $2014$ on the number line, without ever landing on a lily pad containing a snake. What is $p^{1 / 2014}$? Express your answer as a decimal number. If $C$ is the actual answer to this question and $A$ is your answer, then your score on this problem is $\lceil\max \{25(1-20|C-A|), 0\}\rceil$.