# Probability of having boy

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

The proportion of boys to girls is 1 : 1        (Shocked?!!!!)

EXPLANATION-
This is 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.

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. This is correct.

I’d like to add that there are a lot of in-depth probabilistic proofs for this solution, but a solution can be found more easily with common sense (and a little statistics knowledge).

This is an example of the geometric distribution, where X is a random variable denoting how many times a family fails (has a girl) before they see a success (has a boy).

The expected number of boys in each family is 1, since you know that each family continues having children only until they have a boy.

The expected number of girls in each family is the expected value of X, which for the geometric distribution is (1-p)/p = 1, where p=0.5

A very challenging problem, none the less!!

on 18th October 2016.

sorry,mathematically it is not correct. according to you probability of two girls is (1/2*1/2) and so on.
NO IT IS INCORRECT BECAUSE THE EVENT OF HAVING 2 GIRLS IS INDEPENDENT OF EACH OTHER SO IT CAN’T BE 1/4.
SAME IS THE LOGIC FOR 3 GIRLS, 4 GIRLS. KINDLY REPLY ON THIS FACT.

Hi aviana,

Event of getting 2 girls may be independent of each other, but here we are talking about probability of occurring that event.

The possible outcomes are: (B for boy and G for girl)
1: B B
2: B G
3: G B
4: G G

Here, probability of getting 2 girls is 1 out of 4 events.

Thanks.

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 reproducing & 512 couples with Girls take another chance.
There will be 256 boys & 256 girls in second chance…..(1:1)
256 Couples with Boys stop reproducing & 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 stop reproducing & 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.

The answer is given in the question:
Assuming probability of having a boy or a girl is the same, then after a long time, there will be equal numbers of boys and girls.

Why should anyone think that decisions about when to start or stop having children has any effect on their gender?