Question

The Smith family was one of the first to come to the U.S. They had 7 children. Assuming that the probability of a child being a girl is 0.5, find the probability that the Smith family had:
at least 6 girls?

at most 4 girls?

Answer 1

a)

Here, n = 7, p = 0.5, (1 - p) = 0.5 and x = 6
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 6).
P(X >= 6) = (7C6 * 0.5^6 * 0.5^1) + (7C7 * 0.5^7 * 0.5^0)
P(X >= 6) = 0.0547 + 0.0078
P(X >= 6) = 0.0625

b)

Here, n = 7, p = 0.5, (1 - p) = 0.5 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 4).
P(X <= 4) = (7C0 * 0.5^0 * 0.5^7) + (7C1 * 0.5^1 * 0.5^6) + (7C2 * 0.5^2 * 0.5^5) + (7C3 * 0.5^3 * 0.5^4) + (7C4 * 0.5^4 * 0.5^3)
P(X <= 4) = 0.0078 + 0.0547 + 0.1641 + 0.2734 + 0.2734
P(X <= 4) = 0.7734