A primitive village follows a strange custom. They have babies only to have a boy. Whenever a boy is born in a family, they stop having babies and whenever a girl is born, they have a baby again and again till a boy is born.
The probability of having a boy is same as the probability of having a girl. What will be the proportion of boys to girls in the village after some time?
Solution:
1:1 (approx)
Explanation:
We know that the probability of having a boy or having a girl is same and thus, half of the couples will stop after having a boy child. Half of the others will have a girl and will have a baby again. Out of those half of the couples, half will have a boy and will stop and half will have a girl again. This will keep on going like this.
Now, if there are X number of couples, there will be X boys.
1/2 have a boy and stop: 0 girls
1/4 have a girl, then a boy: X/4 girls
1/8 have 2 girls, then a boy: 2*X/8 girls
1/16 have 3 girls, then a boy: 3*X/16 girls
1/32 have 4 girls, then a boy: 4*X/32 girls
…
Total: X boys and
1X 2X 3X 4X
– + – + – + — +… = ~X
Therefore, the proportion of boys to girls will be extremely close to 1:1