Fog Creek Software
Discussion Board




More probability

Girls and Boys 
It is known that a certain family has two children, at least one of whom is a boy. What is the probability that there is a girl child in that family?

Deepak pant(STM)
Saturday, June 18, 2005

BB, BG, GB

2/3 ?

j
Saturday, June 18, 2005

Wrong... the answer should be 1/2. It is just a question whether the other child is girl or boy.

Ketan Mukadam
Saturday, June 18, 2005

Ketan Mukadam, i think you are wrong.You are not taking into consideration that if the first child is a girl than the second child had to have a Boy!!!.
there is no way to have Two girls (as given in question!!!).

hence the correct answer will be 2/3 not 1/2

Deepak Pant

Deepak Pant
Saturday, June 18, 2005

Then I think the question you framed in a wrong way. It does not indicate that you need to consider 1st child and 2nd child. If atleast one out of two children is boy, then automatically, i will draw the sample space as the various combinations possible for 'other' child. I should not be thinking about the "order".

Ketan Mukadam
Sunday, June 19, 2005

If at least one out of two children is boy, then the only thing you can automatically draw is that girl girl is eliminated.  Any other permutations still remain, which are boy boy, boy girl, and girl boy.  Two out of those three have the other child as girl.

j
Sunday, June 19, 2005

That is my whole point... do you conside "boy girl" and "girl boy" different... Ain't they same.... "one boy" and "one girl"

For example, if the question had been something like "what is the probability of having an elder sister in the family?" then I need to consider whether the first (elder) child is girl or boy.... but unless i need to consider "order", i would stick to answer 1/2

Ketan Mukadam
Monday, June 20, 2005

Ketan, you fell victim to the well known fallacy :)

Try to look at the situation from the different angle. If you consider all the families with two kids, there should be equal amount of all four cases present: bb, bg, gb, and gg. Now, the condition imposed by the question ("at least one kid is a boy") is equivalent of removing all the families with the last case (gg) from the pool. Do you see it now?

It is obvious now that the remaining three cases (bb, bg, and gb) are still equiprobable, as they were before shrinking the pool. A girl is present in two of them. Thus, the sought probability should be 2/3.

Dmitri Papichev
Monday, June 20, 2005

Sorry guys, it seems i am the one who is not able to grasp this simple probability question. [Am i loosing common sense??]

Dmitri,

Consider a city where all the families give birth to only to twins (children of same age). Now consider two families both have one girl child and one boy child. Then how do you classify these two families, do they come under your "BG" category or "GB" category.

So when i say that certain families have two children, then what are the possible combinations?

1) Both are boys
2) Both are girls
3) One is boy and other is girl (how can BG different from GB unless i mention (age) thatis elder is girl and younger is boy or the other way round)

And when the constraint of atleast one boy is applied, the (2) above is removed. So we have only possibilities of (1) and (3).

--------------------------------------------------------------------------

I would not like to extend this discussion, since I assume that there is something amiss in the question or I am reading too little :-(

Sorry for streching it too far.....

Thanks for replies

Ketan Mukadam
Tuesday, June 21, 2005

> 1) Both are boys
> 2) Both are girls
> 3) One is boy and other is girl

If you want to merge BG and GB into a single catagory (3), then just remember that (3) has twice as many families as in (1).

Lets say I have two fair coins.  I flip the coins 4000 times and record the result as head head, tail tail, or one head one tail.  Then roughly I will get 1000 head head, 1000 tail tail, and 2000 one head one tail.

If you have doubts, write a simulation program to flip coins.

j
Tuesday, June 21, 2005

*  Recent Topics

*  Fog Creek Home