At least 220 heads in 400 flips (normal approximation)

Question

A fair coin is flipped 400 times. Let X be the number of heads. Using the normal approximation, estimate P(X \ge 220).

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For large n, a binomial 	ext{Binomial}(n,p) is well approximated by a normal with the same mean and variance: 	ext{mean} = np, \qquad 	ext{variance} = np(1-p). Here np = 400 	imes 0.5 = 200 and np(1-p) = 400 	imes 0.5 	imes 0.5 = 100, so X \approx N(200, 100) with standard deviation 10. Convert 220 to a z-score (how many standard deviations above the mean): z = \frac{220 - 200}{10} = 2.0. The chance a standard normal exceeds 2 is P(Z \ge 2) \approx 0.0228, about 2.3%. (A continuity correction uses 219.5 instead of 220, giving z = 1.95 and about 0.0256; both are fine at this level.) Take…

At least 220 heads in 400 flips (normal approximation)

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