Sex

[factor-fiction]

Following on from the entertaining Dice and Socks threads earlier this week I've now found the problem about the sex of children.

One version goes: "You know that Mr. Smith has two children and that at least one of them is a boy. What is the probability that both children are boys?"

Thoughts please?

[ToraToraTora]

sorry this is obvious, evens! someone tell me why this is not so?

[factor-fiction]

Why is it not evens? Well it's not evens according to a lot of articles by learned folk- including someone with a PhD- have said it's 1/3 and produced detailed explanations

[factor-fiction]

Sorry, some text didn't appear there again. Time for a new keyboard, maybe

[sunny-dave]

Because, TTT, a boy/girl combination is twice as likely as a boy/boy one - therefore you are twice as likely to see a girl as the hidden sibling.

the boy/boy is a one in three chance.

[-Talbot-]

I don't appreciate threads with titles like this one.

[sunny-dave]

I love this stuff - you can waste hours and make your brain hurt :)

[ToraToraTora]

you have n kids and all but n minus 1 are of known sex so the only one that matters is the one that is not known all the others are irrelevant, and that can be a boy or a girl, QED! so I don't get it!

[pixie373]

Nor do I. If it can be BG OR GB, surely it can also be B1B2 OR B2B1? The boys can be born in either order too.

[sunny-dave]

I agree for a single, freestanding, child it's 50/50

BUT ... the others are not irrelevant - that's the pitfall.

Assuming a 50/50 chance of boy/girl it really does matter what the sex of the others in the sample is :)

[woofgang]

I agree with TTT knowing the gender of one child or 97 thousand children is irrelevant because the gender of the previous child, like toe coin toss has no real effect. Of the four options, GG is ruled out and, because we don't care which child is older, GB and BG are one option not two. This reduces the options to BB or BG or a one in two chance.

[sunny-dave]

A different explanation :

If you have 1024 families with two children then (approximately)

256 will have two boys

256 will have two girls

512 will have a boy and a girl.

That's just simple probabilities that we can all agree on?


If we select our 'test' family at random (excluding the girl/girl group since we know that one is a boy) we will have 256 chances of picking a boy/boy family and 512 chances of picking a boy/girl family.


Which gives a one in three chance of boy/boy.

[ludwig]

I think the trick is in the phrase 'at least one of them is a boy'.
It's actually irrelevant to the problem, and lures you down an illogical path, that seems logical.

The question is really 'A man has two children. What is the probablility that both children are boys'?

The answer to that is 1/3.

[ToraToraTora]

right, the penny is dropping I think I get it, from the outset it could be GB/BG/GG/BB by getting rid of those with B in the second column we have 3 left 2 of which have G in col 1. Eureka, nice problem, similar to the old monty norman game show problem, that gets people thinking in the wrong direction too!

[factor-fiction]

So dave:
Suppose Mrs Jones has two children. The first is a boy. Are you saying the probability of the second being a boy is only 1 in 3?

[Old_Geezer]

I believe it is a third, but I confess it had me going for a while making assumptions that proved to be wrong.

At least one is a boy, doesn't mean you can dismiss that person's gender. It means a) they don't have two girls and b) They either had boy then another or another then boy.

Boy then boy is one option. Boy then girl is a second option. Girl then boy is a third option.

Only 1 in 3 give both boys.

[factor-fiction]

> "The question is really 'A man has two children. What is the probablility that both children are boys'? The answer to that is 1/3. "

No, surely that's a 1/4 chance.

[sunny-dave]

If you mean 'first born' then yes, factor, the second born is a 50/50.

But that's not what the question says ...

[Old_Geezer]

That is not what the question really is. Make that assumption and you'll come to an incorrect answer.

[-Talbot-]

Please hurry and settle this, I have been away from home for 5 weeks, and this keeps popping up in latest posts!

[pixie373]

It is 1/4. 1/2 x 1/2