Pure Dice
---- ----

Olvk is very concerned by bias on dice, and wants to
make an unbiased 6-sided dice.  He does this by taking
an N sided dice and throwing it 3 times.  If the 3
values thrown are different, they can come out in one of
6 orderings, and each ordering has the same probability.
By assigning a number from 1 through 6 to each of the 6
orderings, Olvk effectively has an unbiased 6-sided
dice.

For example, if S represents the smallest of three
different values, M the middle value, and L the largest
value, then the 6 orderings might be assigned values by

	SML	1
	SLM	2
	MSL	3
	MLS	4
	LSM	5
	LMS	6

One problem is that the three values may not all be
different: if N were 6, one might throw 115 or even 111.
In this case Olvk choses to ignore the three values he
just threw, and throw 3 more times.  If we call a set of
3 throws a `round', Olvk keeps throwing rounds until he
gets 3 different values.  Olvk must throw M rounds to
get an 1 unbiased throw if the first M-1 rounds each
have 2 or 3 equal values and the M'th round does not.

You are asked to compute the probability that Olvk will
need exactly M rounds to get 1 unbiased value, given N
and the probabilities that each of the N faces of the
biased dice will be thrown.


Input
-----

For each case, 1 line containing

	M N p1 p2 ... pN

where M and N are as above and p1, p2, ..., pN are the N
probabilities that each of the N faces of the biased die
will be thrown.  1 <= M <= 100.  4 <= N <= 20.
0.0 <= pI <= 1.0, for I from 1 through N.  The sum of
the pI's is 1.0.

Input ends with an end of file.


Output
------

One line per case containing the probability that it
will take Olvk exactly M rounds to get his first un-
biased value.  This probability must have exactly 6
decimal places.


Example Input
------- -----

1 4 0.2 0.2 0.2 0.4
3 4 0.2 0.2 0.2 0.4

Example Output
------- ------

0.336000
0.148141


File:	   puredie.txt
Author:	   Bob Walton <walton@deas.harvard.edu>
Date:	   Sat Oct 18 22:07:29 EDT 2003

The authors have placed this file in the public domain;
they make no warranty and accept no liability for this
file.

RCS Info (may not be true date or author):

    $Author: walton $
    $Date: 2003/10/19 02:07:56 $
    $RCSfile: puredie.txt,v $
    $Revision: 1.7 $