Planet Cake

In the planet Cake, home of the Master Masao, a casino offers a
particular game. There is an array of probabilities p₁, …, p_(2m + 1)
for some natural number m. At every moment, a coin has probability p_(i)
of landing heads when flipped. If it indeed lands heads, the next time
the probability will be p_(i + 1). Otherwise, the probability will be
p_(i − 1). The initial “state” is m + 1. Before playing, you must decide
a number k between 1 and m + 1. Afterwards, you flip the coin k times.
You win if the total number of times the coin landed heads is an odd
number.

Given the probabilities of a coin, compute the probability of winning a
game assuming an optimal strategy.

Input

Input consists of several cases, each with an odd number n followed by
n probabilities. Assume n < 50.

Output

For every case, print the probability of winning with four digits after
the decimal point. The input cases have no precision issues.

Problem information

Author: Xavier Martínez

Generation: 2026-01-25T10:06:11.034Z

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