The first step for this problem is to map out what the probability that x is even would be:
$ P[X is Even]= \sum\limits_{k=0,even}^N P[x=k]= \sum\limits_{k=0}^N {N \choose k} P^k(1-P)^{N-k}[1+(-1)^k] $ Eq. 1
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Is this true? Won't the probability double because when k is even $ [1+(-1)^k] = 2 $? Shouldn't you put a 1/2 somewhere because of this?? Please advise. Thanks,
Ken Pesyna
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Next we must expand $ (x+y)^n + (x-y)^n $ using the binomial thereom:
$ (x+y)^n + (x-y)^n = \sum\limits_{k=0}^N {N \choose k} y^kx^{N-k} + \sum\limits_{k=0}^N {N \choose k} (-y)^kx^{N-k} $ Eq. 2
This simplifies to:
$ \sum\limits_{k=0}^N {N \choose k} (x)^{N-k}y^k[1+(-1)^k] $ Eq. 3
By comparing equations 1 and 3 you can derive a condensed equation in the form of $ (x+y)^n + (x-y)^n $ where $ x = 1-P $ and $ y=P $ :
$ ((1-P)+P)^n + ((1-P)-P)^n = 1^n + (1-2P)^n $ Eq. 4
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I might be wrong, but 1^n + (1-2P)^n should be greater than 1. If so, P[X is even] > 1, which doesnt look reasonable.
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Note that even though the hint gives away x = 1-P and y = P, we use the binomial theorem to prove the correlation between equations 1 and 3.
