Question

What’s the probability of flipping a coin 25 times and getting between 32% and 50% heads. Please assume that this clearly discrete distribution is, in fact, continuous, and use the normal approximation.

Answer 1

Let X be the random variable denoting the number of heads

obtained after flipping a fair coin 25 times.

Thus, X ~ Bin(50, 0.5).

Here, np = 50 * 0.5 = 25 > 10 and n(1-p) = 25 * 0.5 = 25 > 10

Hence, using normal approximation, E(X) = np = 25 and

Var(X) = np(1-p) = 12.5.

Hence, X ~ N(25, 12.5) i.e. (X - 25)/3.5355 ~ N(0,1)

32% of 25 = 8 and 50% of 25 = 12.5.

Thus, required probability = P(8 < X < 12.5)

=P[(8 - 12.5)/3.5355 < (X - 12.5)/3.5355 < (12.5 - 12.5)/3.5355]

= P[-1.2728 < (X - 12.5)/3.5355 < 0] = (0) - (- 1.2728)

[(.) is the cdf of N(0,1)]

= 0.5 - 0.1015 = 0.3985. (Ans).