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Suppose you flip a weighted coin that is $3$ times more likely to come up heads. What is the probability that, if you flip the coin $3$ times, you will get an even number of heads?

Can someone help me think about this problem without having to explicitly list possibilities in the sample space or event space? For example, if the problem had called for flipping the coin $1000$ times.

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This is an example of the binomial distribution: en.wikipedia.org/wiki/Binomial_distribution –  JavaMan Nov 7 '12 at 3:55
    
Can you elaborate on how exactly this helps? –  user1038665 Nov 7 '12 at 4:25

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up vote 8 down vote accepted

Let's say you flip the coin $n$ times and it has probability $p$ of coming up heads. Then the probability of getting an even number of heads is

$$ \begin{align} \sum_{k\text{ even}}\binom nkp^k(1-p)^{n-k} &= \frac12\left(\sum_k\binom nkp^k(1-p)^{n-k}+\sum_k\binom nk(-p)^k(1-p)^{n-k}\right) \\ &= \frac12\left(1+(1-2p)^n\right)\;. \end{align} $$

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