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A fair coin is flipped $10^6$ times.

What's the probability that the number of heads is at least 499000 and at most 501000?

I'm not sure how to even go about starting this. Does it involve the $Q$ function?

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  • $\begingroup$ Hast thou studied the binomial distribution, by any chance? $\endgroup$ – dfeuer Oct 7 '13 at 2:59
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    $\begingroup$ Look also at the normal approximation to the binomial distribution. Them numbers be large. $\endgroup$ – dfeuer Oct 7 '13 at 3:05
  • $\begingroup$ Thanks. This is mostly what I was looking for. Found a video that explains it pretty well. $\endgroup$ – user2503227 Oct 7 '13 at 3:12
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By the central limit theorem, the distribution may be well approximated by a normal distribution. The mean $\mu = 10^6 (1/2) = 500000$, and the variance is $\sigma^2 = 10^6 (1/2) (1/2)$ so that the standard deviation is $\sigma = 500$. You are then asked the probability of being within $\pm 2$ standard deviations of the mean.

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  • $\begingroup$ Or punch the exact binomial distribution into, say, ghci, and get the exact rational answer quickly enough :P $\endgroup$ – dfeuer Oct 7 '13 at 3:12

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