Probability of having boy

By | December 11, 2017

Probability of having boy : This is a question often asked with referencing different things. Most of the time it is asked with probability of having boys in a country. It’s a very simple probability question in a software interview.

This question might be a little old to be ever asked again but it is a good warm up.

The question goes like,  In a country where everyone wants a boy, each family continues having babies till they have a boy. After a considerable amount of time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same).

 

Take some time to think before looking at the explanation below.

 

Solution & Explanation:

It will always be 1:1 for large number of couples…

Take any number as couples N e.g. 1028.
There will be 512 boys & 512 girls at first delivery….. (1:1)

Couples with Boy stops having child & 512 couples with Girls take another chance.
There will be 256 boys & 256 girls in second chance…..(1:1)

256 Couples with Boys stops having child & 256 Couples with 2 Girls Now will take another chance….
There will be 128 boys & 128 girls in third chance………(1:1)

128 Couples with Boys stops having child & 128 Couples with 3 Girls Now will take another chance….
There will be 64 boys & 64 girls in fourth chance………(1:1)

and ratio will continue till everyone has a boy each………………….then at end there will be N boys &  N-1 girls…
So ratio will be N:N-1 which is 1:1 in case of large sample size of N.

 

Mathematical Explanation: 

Assume there are C number of couples so there would be C boys. The number of girls can be calculated by the following method.

Number of girls = 0*(Probability of 0 girls) + 1*(Probability of 1 girl) + 2*(Probability of 2 girls) + …
Number of girls = 0*(C*1/2) + 1*(C*1/2*1/2) + 2*(C*1/2*1/2*1/2) + …
Number of girls = 0 + C/4 + 2*C/8 + 3*C/16 + …
Number of girls = C
(using mathematical formulas; it becomes apparent if you just sum up the first 4-5 terms)

The proportion of boys to girls is 1 : 1.