Question

If a family has 7 children, what is the probability that they will have at least 5 boys?

Answer 1

This is a direct application of Binomial distribution.

Let X be a number of boys among 7 children.

P(boy) = P(girl) = p = 0.5

Here, X ~ Binomial ( n = 7, p = 0.5)

Probability mass function of X is,

P(X = x) = ⁿC_x p^x (1-p)^n-x

We want to find, P(X >=5)

P(X >= 5)

= P(X = 5) + P(X = 6) + P(X = 7)

= 0.1641 + 0.0547 + 0.0078

= 0.2266

P(X >= 5) = 0.2266

The probability that they will have at least 5 boys is 0.2266

Note:

P(X = 5) = ⁷C₅ * (0.5)⁵ * (1-0.5)^7-5 = 21 * (0.5)⁷ = 0.1641

P(X = 6) = ⁷C₆ * (0.5)⁶ * (1-0.5)^7-6 = 7 * (0.5)⁷ = 0.0547

P(X = 7) = ⁷C₇ * (0.5)⁷ * (1-0.5)^7-7 = 1 * (0.5)⁷ = 0.0078