Probability of a Girl in New Family of 2 Adolescents

[Rogerio]

Suppose boys and girls are born at the same rate.

A new family has just moved to your next door.

Besides the parents, you know there are 2 adolescents, and your sister has already told you at least one of them is a boy.

What is the probability that the other one is a girl?

 

[chroot]

I feel like I must be missing the point of this... the probability of giving birth to a girl is the same as the probability of giving birth to a boy, as you said in the first line. The births are independent, so there is a 50% probability that ANY unknown child is a girl.

Perhaps the "at least one of them is a boy" part is supposed to be a red herring or something? I don't get it.

- Warren

 

[NateTG]

I feel like I must be missing the point of this... the probability of giving birth to a girl is the same as the probability of giving birth to a boy, as you said in the first line. The births are independent, so there is a 50% probability that ANY unknown child is a girl.

Perhaps the "at least one of them is a boy" part is supposed to be a red herring or something? I don't get it.

There are four possibilities:
BB
BG
GB
GG
The existence of a son eliminates the bottom possibility.
Therefore the odds of the familiy having a daughter is $$\frac{2}{3}$$.

 

[chroot]

Er uh... oh yeah.

- Warren

 

[BobG]

Select to see answer:

50%

There are four possible 2 kid combinations:

GG
GB
BG
BB

The boy the daughter saw could be the boy from the GB family, in which case he has an older sister.

The boy could be the boy from the BG family, in which case he has a younger sister.

The boy could be the oldest boy from the BB family, in which case he has a younger brother.

The boy could be the youngest boy from the BB family, in which case he has an older borther.

 

[Gokul43201]

There are four possibilities:
BB
BG
GB
GG
The existence of a son eliminates the bottom possibility.
Therefore the odds of the familiy having a daughter is $$\frac{2}{3}$$.

Given that one boy has been seen, the odds for BB is twice as great as the odds for BG (or GB). ie : BB, GB, and BG are no longer equiprobable.

So, it is 50% after all, isn't it ?

Alternatively, P(G|B) = P(G^B)/P(B) = 0.25/0.5 = 0.5

 

[NateTG]

Given that one boy has been seen, the odds for BB is twice as great as the odds for BG (or GB). ie : BB, GB, and BG are no longer equiprobable.

So, it is 50% after all, isn't it ?

Alternatively, P(G|B) = P(G^B)/P(B) = 0.25/0.5 = 0.5

It's a bit ambiguous. If it's that the sister saw a child at random, and it was a boy, then the odds are 50%, if the information is that they have at least one male child, then it's 2/3.

 

[BobG]

It's all in the semantics of the question.

2/3 of all families having boys also have a girl.

That's not the same as saying each boy has a 2/3 chance of having a sister. The chances of any given boy's sibling being boy or girl is equal.

In this case, Rogerio asked what is the probability of the 'other' one being a girl, not "What are the chances that the family also has a girl?"

In other words, NateTG gave a correct answer, but not the answer to Rogerio's question.

 

[BobG]

This question appeared in "Ask Marilyn" in Parade Magazine (Marilyn is Marilyn Mach Vos Savant, who holds the world's record for highest IQ score).

Marilyn agreed with NateTG. Wait, I mean she proved the NateTG's answer was wrong. No, wait, I mean she actually did both. She agreed with Nate's answer in the magazine, but disagreed with his answer in her book.

http://www.geocities.com/SiliconValley/Circuit/1308/mvsm.html

Maybe the fact that the magazines addressed humans and the book addressed pancakes makes a difference. After all, as Clem McCarthy replied to Bill Stern, "Remember, Bill, you can't lateral a horse."

FYI - Clem McCarthy, radio horse racing sportscaster, called the wrong winner of the 1947 Preakness when he lost track of two horses, both wearing red silks, as the rounded the final bend. Bill Stern asked him "But couldn't you see the numbers as they headed down the home stretch?"

Clem was referring to a radio football sportscast, where Bill Stern suddenly realized he was calling the wrong player streaking down the field alone and improvised "And Larrimer laterals to Hanratty" so the listeners wouldn't realize the mistake when they read the box score in the newspapers the next morning.

 

[Gokul43201]

No matter, it was quite clever (not to mention, funny) to make up a lateral to cover a screw-up. Quick thinking, wot ?

 

[TenaliRaman]

ah ! Monty Python !
err Monty Hall doesn't sound exciting!