Puzzle | The Boy Preference Ratio Riddle

In a country, every family continues to have children until they have a boy, after which they stop having more children. Assuming the probability of having a boy or a girl is equal (50%), what is the expected ratio of boys to girls in the overall population?

Check if you were right - full answer with solution below.

Solution:

Assumptions: Each child born has an equal probability of being a boy or a girl (i.e., 50%). The gender of each child is independent of the previous births. To solve the problem, we calculate the expected number of girls born before a boy appears in each family.

Let NG be the expected no. of girls before a boy is born

Let p be the probability that a child is girl and (1-p)
be probability that a child is boy.

NG can be written as sum of following infinite series.

NG = 0*(1-p) + 1*p*(1-p) + 2*p*p*(1-p) + 3*p*p*p*(1-p) + 4*p*p*p*p*(1-p) +.....

Putting p = 1/2 in above formula.

NG = ( 1-1/2) .1/2(1-1/2)2 = 1/2. 1/2/ ( 1/2)2= 1/2. 1/2/ 1/4= 1/2.2= 1

NG = 1

So,

• Expected number of boys per family = 1
• Expected number of girls per family = 1
• Therefore, in the whole country:

⁛ Ratio of boys to girls=1/1​=1:1​