# Probability of getting 499000–501000 heads if a fair coin is flipped $10^6$ times

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?

• Hast thou studied the binomial distribution, by any chance? – dfeuer Oct 7 '13 at 2:59
• Look also at the normal approximation to the binomial distribution. Them numbers be large. – dfeuer Oct 7 '13 at 3:05
• Thanks. This is mostly what I was looking for. Found a video that explains it pretty well. – user2503227 Oct 7 '13 at 3:12

## 1 Answer

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.

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