The King’s Party

Now, this one took me quite a while to figure out.

You are a knight (say, Knight of the North) who is attending the king’s party. You meet a guest, who tells you that he has two children. During the conversation, you also learn that one of the children is a girl. You do not inquire about the other child (as you were discussing far more important stuff like politics, diplomacy, economy and women). Now, after the party the king asks you to send that particular guest gifts for his children. You say to yourself, “I would be able to send more suitable gifts if I know the gender of both the children. But since I do not know the gender of both, let me guess the most probable gender of the other child, and send gifts accordingly.”

If the guest has two children, and you know that one of them is a girl, what is the probability that the other child is a boy?

You would say, ½. This is like flip of a coin, right? If we get heads once, the probability of getting heads again when the coin if flipped the second time, is ½, since the two events are independent.  Now, there is another way of looking at it, which leads to another result. I would suggest you give some thought to this.

Another knight, say, Knight of the South, suggested a different method. He says: All men who have two children fall in one of the following four cases:

	Older Child	Younger Child
Case 1	Boy	Girl
Case 2	Boy	Boy
Case 3	Girl	Girl
Case 4	Girl	Boy

You do not dispute this, do you? Now, all the four cases given above are equally likely. Agreed?

Now, if one of the children is a girl, the man falls in cases 1, 3 or 4. In two of these three cases, the other child is a boy (Case 1, Case 4). Hence, the probability of one of the children being a boy, given that one child is a girl, is 2/3.

The logic is flawless. You have to agree. Go over it once again, if you wish.

This is a classic puzzle, known as ‘boy or girl paradox’, first published (probably) by Martin Gardner in Scientific American.

But this is not the end of the story. There is more to it. Much more. Now, suppose, a friend of yours gives you an extra piece of information.

“I know the girl the guest was talking about that day, you know! I know that she was born on a Tuesday”.

 “Oh was she?” say you, “but how does that matter to me? Does it, O Knight of the South, the great warrior and the leader of the wise men?”

“It does, O Knight of the North, the master of immense wealth, it changes the entire calculation”, says the Knight of the South.

“Just how, my friend? Please enlighten me on this. I do not doubt your wisdom, oh how can I, O leader of the wise men, but this does not seem logical that this new information, that the girl was born on a Tuesday, may change anything in our guess about the gender of the other child.”, you say.

“Listen carefully, my friend”, says the knight of the South, “Following is the logic on which I base my argument”. Then, the knight explains the following.

(Click here to see the detailed calculation)

It so turns out that if we know that the girl was born on a Tuesday, the chance that the other child is a boy is 0.52, which is very close to ½ . Intuitively, it will get closer to ½, if we know the date of birth of the girl and get even closer if we also know her name! If you know the girl, or meet her, the probability that she has a brother goes to exactly ½. (And this last statement actually makes sense intuitively).

So, this is the paradox. How can an irrelevant piece of information make an impact on probability? Perplexing, isn’t it?