Puzzle 44 | Girl or Boy

The Boy or Girl problem consists of a bunch of questions in probability which has kept the mathematicians on discussion about its possibilities.  But different from them, here is one puzzle for you with no ambiguity.

Suppose that I have a friend whose name is Ankur. Ankur has his elder brother who got married 10 years before. But Ankur being a silly guy has the habit of forgetting things. This summer he is going to meet his brother after 10 years, and he has planned to purchase some clothes for his brother’s kids as a token of love. Each and every time he faces a problem he comes to me, so this time also, he came to seek help, and I have not left him empty handed. I told him that as far as I can recall, his brother has two offspring’s and one of the boy was born on Tuesday. Now, it’s your turn to help him.  What is the probability that Ankur’s brother has two boys, with at least one of them born on Tuesday? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

Solution: If you think the answer should be 1/2, you would be wrong. If you knew which child was a boy (say, the younger one), you would be closer to the truth. But since the boy could be either the younger or the older child, the analysis is more subtle. The answer would be 13/27.

There can be four cases, i.e. either his brother has two boys, or one boy(elder) and one girl, one girl(elder) and one boy, or both of them are girls. There are 7 equally likely possibilities for any one of the children of taking birth on any day for all these four cases. The total cases where both the children are boys and at least one of them is born on Tuesday is 7×2-1 (as there is repetition of two boys two times). The total cases when one of the child is a boy born on Tuesday is 7×4- 1. Therefore, using conditional probability, the probability of both children being boys, with at least one of them born on Tuesday, given that one is a boy born on Tuesday, is equal to 13/27. The things can be made more clear from the image shown below.

This article is contributed by Praveer Satyam. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above