Question

The probability of birth of a girl is 0.5. A couple wants to have children until they have at least two girls in their families.
What is the probability that the family will have four children?

A. 1/16

B. 21/32

C. 7/16

D. 11/32

Answer 1

n = 4

p = 0.5

q = 1 – p = 1 – 0.5 = 0.5

Using Binomial distribution

P(X=r) = ⁿC_r p^r (1 - p)^{(n - r)}

P( at least two girls):

Here P( r   2 ) = P( r = 2) + P( r = 3) + P(r = 4)

=  ⁴C₂(1/2)² (1/2)² + ⁴C₃ (1/2)³ (1/2)¹ + ⁴C₄ (1/2)⁴ (1/2)⁰

= 6 (1 / 16) + 4 (1 / 16) + (1 / 16)

= 11 / 16

We can also derive this relation using binary combinations as:

B = Boy

G = Girl

	B	B	B	B
	B	B	B	G
	B	B	G	B
	B	B	G	G
	B	G	B	B
	B	G	B	G
	B	G	G	B
	B	G	G	G
	G	B	B	B
	G	B	B	G
	G	B	G	B
	G	B	G	G
	G	G	B	B
	G	G	B	G
	G	G	G	B
	G	G	G	G

There are total 11 cases out of 16 which have 2 or more than ( At least 2) girls.

Thus, 11/ 16.