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?

[ludwig]

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

I make it 3 possibilities :-

2 boys
2 girls
1 of each

Unless there's some other combination I'm missing?

[woofgang]

But Ludwig, we KNOW one is a boy so that rules out GG BG and GB are one option, not two so the options are BB or BG, which takes us back to one in two.

[factor-fiction]

Boy then girl,
girl then boy
boy then boy
girl then girl

It's a bit like tossing two coins- the chances of two heads is 1/4 not 1/3. The chances of one of each is 2/4

[jim360]

Once again, the errors people are making is that they're simplifying the problem too early. In probability you should always start by finding all possible outcomes and their probabilities, and only once you've done so can you consider the specific question. Here we have, from the outset, four equally likely outcomes (assuming boys or girls are equally likely) of GG, GB, BG, BB, of which one (GG) is excluded and one (BB) is what we're asked for. So it's 1/3 not 1/2 as GB and BG are two separate outcomes with the same probability.

[Old_Geezer]

Not after you have removed one of the four as impossible.

[sunny-dave]

BG and GB are not the same ... it can be a younger boy and an older girl or a younger girl and an older boy - two distinct possibilities.

[ToraToraTora]

but there are 2 ways of having one of each:
BB/GB/BG/GG - are the possibilities hence 1 in 4

[pixie373]

Exactly what I said, woofgang. We seem to be taking into account the birth order of boy /girl, but not the other 2 options.

[ToraToraTora]

I mean 1 in 4 for having 2 boys.

[ludwig]

The order of births is a complete red herring.

We are being asked what children he actually HAS, not what the next one to be born might be. As dave said, the coin has already been tossed.

There are only 3 combinations as I said above. Of those, only 1 of them is boy/boy.

[ToraToraTora]

ie 1/2 X 1/2 = 1/4!

[factor-fiction]

Oh dear.

[Prudie]

I lost the internet during posting so my answer is a bit late but earlier I was going to say:

Because tossing 2 coins has 4 outcomes and once told 1 of them is not a result then yes there are 1 in 3 possibilities.

Two children, 1 is a boy - not told if it's older, younger, whatever ie one coin is already tossed - only 2 possible outcomes for the other.

[ToraToraTora]

the chance of 2 the same from the outset is 1/4

[sunny-dave]

Pixie - if you have two boys then you always have a younger boy and an older boy ... one option

But, if you have one of each, then it can be a younger boy and an older girl or a younger girl and an older boy - two distinct options.

[Old_Geezer]

You have the right answer but for the wrong reasons ludwig.

GG is impossible and can be removed.
1 of each, is 2 options, BG and GB.

Still 3 options though.

[Old_Geezer]

Yes TTT. But that isn't the question.

[sunny-dave]

Prudie - you don't know whether you have the result of the first or second event ... it does matter.

[ludwig]

Ok, what are the 4 sex combinations for two children.?

1) 2 boys
2) 2 girls
3) 1 of each
4) .................please fill in the 4th one here, because whatever it is, I'm interested to know.

[ToraToraTora]

yes OG i was referring to the later question of the chance of 2 boys from the outset from ludwig, that is 1/4. The original question answer was 1/3.