I - Coins

Problem Statement

Let $\mathrm{NN}$ be a positive odd number.

There are $\mathrm{NN}$ coins, numbered $\mathrm{1,2,\dots ,N1,2,\dots ,N}$. For each $\mathrm{ii}$ ($\mathrm{1\le i\le N1\le i\le N}$), when Coin $\mathrm{ii}$ is tossed, it comes up heads with probability ${\mathrm{pipi}}^{}$ and tails with probability $\mathrm{1-pi1-pi}$.

Taro has tossed all the $\mathrm{NN}$ coins. Find the probability of having more heads than tails.

Constraints

• $\mathrm{NN}$ is an odd number.
• $\mathrm{1\le N\le 29991\le N\le 2999}$
• ${\mathrm{pipi}}^{}$ is a real number and has two decimal places.
• $\mathrm{0<pi<10}$

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Input

Input is given from Standard Input in the following format:

$\mathrm{NN}$
${\mathrm{p1p1}}^{}$ ${\mathrm{p2p2}}^{}$ $\dots \dots$ ${\mathrm{pNpN}}^{}$

Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than ${\mathrm{10-910-9}}^{}$.

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Sample Input 1 Copy

Copy

3
0.30 0.60 0.80

Sample Output 1 Copy

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0.612

The probability of each case where we have more heads than tails is as follows:

• The probability of having $(Coin1,Coin2,Coin3)=(Head,Head,Head)(Coin1,Coin2,Coin3)=(Head,Head,Head)$ is $\mathrm{0.3\times 0.6\times 0.8=0.1440.3\times 0.6\times 0.8=0.144}$;
• The probability of having $(Coin1,Coin2,Coin3)=(Tail,Head,Head)(Coin1,Coin2,Coin3)=(Tail,Head,Head)$ is $\mathrm{0.7\times 0.6\times 0.8=0.3360.7\times 0.6\times 0.8=0.336}$;
• The probability of having $(Coin1,Coin2,Coin3)=(Head,Tail,Head)(Coin1,Coin2,Coin3)=(Head,Tail,Head)$ is $\mathrm{0.3\times 0.4\times 0.8=0.0960.3\times 0.4\times 0.8=0.096}$;
• The probability of having $(Coin1,Coin2,Coin3)=(Head,Head,Tail)(Coin1,Coin2,Coin3)=(Head,Head,Tail)$ is $\mathrm{0.3\times 0.6\times 0.2=0.0360.3\times 0.6\times 0.2=0.036}$.

Thus, the probability of having more heads than tails is $\mathrm{0.144+0.336+0.096+0.036=0.6120.144+0.336+0.096+0.036=0.612}$.

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Sample Input 2 Copy

Copy

1
0.50

Sample Output 2 Copy

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0.5

Outputs such as 0.500, 0.500000001 and 0.499999999 are also considered correct.

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Sample Input 3 Copy

Copy

5
0.42 0.01 0.42 0.99 0.42

Sample Output 3 Copy

Copy

0.3821815872

 

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