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Omni-MATH / Consider a number line, with a lily pad placed at each integer point. A frog is…

Problem

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problem

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 nn on the number line, it can jump directly to either point n+2n+2 or point n+3n+3 on the number line. Each of the lily pads at the points 1,⋯,20131, \cdots, 2013 on the number line has, independently and with probability 1/21 / 2, a snake. Let pp 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 p1/2014p^{1 / 2014}? Express your answer as a decimal number.
Plain-text mathematical notation (without MathML)
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,⋯,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.
Original LaTeX notation
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.

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Source and history

task_id
401633a3-662f-5181-ad0b-c77f7ab2f17f
task_key
test--401633a3-662f-5181-ad0b-c77f7ab2f17f
task_revision_id
1
config / split
/ test

Official source

initial import