Boy or Girl?

[Eduk]

A woman is stopped on the street and asked two questions.

1. Do you have two children.
2. Is at least one of those children a boy.

She answers yes and yes.

What is the probability that both of her children are boys?

I do have a point in mind when asking this question. I just want to see some replies first as I wonder if the answers will be illuminating (or not).

[Killosopher]

A woman is stopped on the street and asked two questions.

1. Do you have two children.
2. Is at least one of those children a boy.

She answers yes and yes.

What is the probability that both of her children are boys?

I'll answer this in 2 ways.

1. Let B and G = boy and girl respectively
S = {BB, BG, GB, GG}
A = first is B
C= both are boys
A= {BB,GG}
A n B = {HH}
P (A n B) = 1/4
P(A)= 1/2
P (B\A) = P (A n B)/P(A)= 1/4/1/2 = 0.25/0.5 =0.5

2. An individual can produce about 8,388,608 Ov or Sp. Since the union of Ov and Sp is random. A couple can therefore, reproduce about 8,388,608 x 8,388,608 = 70,368,774,177,664 different children. So the probability that the second is also a boy (an XY instead of an XX) would be 1/ 70,368,774,177,664 = 0.0000000000000014. Or may be its 1/8,388,608 = 0.00000012 since the father is responsible for the Y baring Sp.

The first answer is probably right. The second might be a bit silly. Orr... there is a probability that both are wrong, take you pick.

[Killosopher]

A n B = {HH}

*A n B = {BB}*
Don't mind the typo.

[LuckyR]

A woman is stopped on the street and asked two questions.

1. Do you have two children.
2. Is at least one of those children a boy.

She answers yes and yes.

What is the probability that both of her children are boys?

I do have a point in mind when asking this question. I just want to see some replies first as I wonder if the answers will be illuminating (or not).

The common sense way of thinking about this question is: OK if I have 2 children and one is a boy then there is a 50% chance that the other one is a boy (and a 50% chance that the other one is a girl) therefore the answer is 50%

However, the better answer is that there are four possible outcomes of 2 children: 1 of boy/boy, 1 of girl/girl and 2 of boy/girl (the other is girl/boy). If you eliminate girl/girl, then the chance of boy/boy is 33.3% (the 66.7% is boy/girl and girl/boy)

The best answer is that while the above is true, that boys and girls are not born at a 50/50 ratio so since there is a 51% chance of having a boy (and a 49% chance of having a girl) the chance of boy/boy is 34.2%

[Killosopher]

The common sense way of thinking about this question is: OK if I have 2 children and one is a boy then there is a 50% chance that the other one is a boy (and a 50% chance that the other one is a girl) therefore the answer is 50%

However, the better answer is that there are four possible outcomes of 2 children: 1 of boy/boy, 1 of girl/girl and 2 of boy/girl (the other is girl/boy). If you eliminate girl/girl, then the chance of boy/boy is 33.3% (the 66.7% is boy/girl and girl/boy)

The best answer is that while the above is true, that boys and girls are not born at a 50/50 ratio so since there is a 51% chance of having a boy (and a 49% chance of having a girl) the chance of boy/boy is 34.2%

Lucky, the woman can only give birth twice for 2 kids. The first is already given as B, which automatically eliminates the possibility of {GG,GB}. This leaves us with 2 plausible outcomes {BB, BG} and out of these the probability of the second child also being a boy is 1/2 = 0.5 = 0.5 x 100 = 50%

Unfortunately, I can't find any thing online to back me up so I'm probably be wrong.

More typos,

A= {BB,GG}
A n B = {HH}

*A = {BB,BG}*
*A n B = {BB}*

take you pick.

Take your pick*

[Eduk]

Yes I should have said let's just imagine a perfect split between male and female. Exact real world ratios of man to woman can be ignored.

[Killosopher]

Oops, I thought the first born was a boy. My bad, I totally misread the question. I second lucky's answer.

[Gertie]

Killosopher I thought you were right first time.

If the question is asking for my answer from my pov, knowing what Eduk told us, gotta be 50/50ish innit? (excluding stuff like transgender, hermaphrodite, or somesuch). The other kid is either a boy or a girl. Doesn't matter which order they were born.

Another potential catch is she says yes she has two children, but not only two, maybe she has six altogether. But the question asks 'both', so that would be naughty.

[Fooloso4]

The sex of a child is not analogous to the flip of a coin or the probability of a marble being either black or white. First, the child’s sex may be ambiguous. Second, the ration of boys to girls is not 50/50. The birth rate is slighter higher for males: 1 female to 1.7 male. Third, there may be an inherited tendency passed from fathers making it more likely to have sons or daughters.
sciencedaily.com/releases/2008/12/08121 ... 121835.htm

And fourth, there may be a geographical factor and an age factor.
pewresearch.org/fact-tank/2013/09/24/th ... -you-live/

If we were talking about coin flips or marbles the likelihood is 50/50 that the second would be the same as the first. The results of the first have no influence on the probability of the second.
If we are talking the gender of a child, all of the above would have to be factored in.

[Eduk]

Just to reiterate in this case we assume a 50/50 spilt like a perfect coin toss.
Also she has two children. Not at least two children.

There is no intended ambiguity, any ambiguity is down to my failure to pose the question properly.

Which I am happy to resolve as we go.

[Killosopher]

Killosopher I thought you were right first time.

If the question is asking for my answer from my pov, knowing what Eduk told us, gotta be 50/50ish innit? (excluding stuff like transgender, hermaphrodite, or somesuch). The other kid is either a boy or a girl. Doesn't matter which order they were born.

Gertie, the birth order matters because had the first been a boy it would have allowed me to eliminate 2 of the sample spaces {GG} and {GB}. But since the birth order is not known I can only eliminate 1 sample space {GG} as both can't be girls since one is already a boy, which leaves me with 3 possible out comes {BG}, {BB} and {GB} making the probability 1/3 = 0.33... = 33.3% as Lucky pointed out in his/her 2nd paragraph.

Another potential catch is she says yes she has two children, but not only two, maybe she has six altogether. But the question asks 'both', so that would be naughty.

Huh? lol

A = first is B
C= both are boys
A= {BB,GG}
A n B = {HH}
P (A n B) = 1/4
P(A)= 1/2
P (B\A) = P (A n B)/P(A)

Please excuse any other errors that I failed to notice.
A = first is a B
C= both are Bs
A= {BB,BG}
A n C = {BB}
P (A n C) = 1/4
P(A)= 1/2
P (C\A) = P (A n C)/P(A)

[Fooloso4]

If the probability of each event is independent as in a coin toss the chance of a boy is equal to the chance of a girl, 50%, 50/100 or ½. We are taught to multiply probabilities, and so it might seem that we should multiply ½ (for the first child) x ½ (for the second child) to arrive at a probability of ¼ or 25%. But since we already know that one child was a boy the probability in that case would be 1 not 1/2. So if we multiply probabilities we would have 1 x ½ and arrive at the probability of the second child being a boy as 50%.

[Sy Borg]

There may be other factors skewing the sex ratio from the intuitive 50/50 split: sciencemag.org/news/2014/12/why-women-s ... ough-times

Bruckner turned to Virpi Lummaa, a biologist at the University of Sheffield in the United Kingdom who had digitized centuries of Finnish church records that documented family histories and recorded other data about every Finn born at those times. The researchers looked at the sex ratios of newborns from 1790 through 1870 and tallied how many males survived infancy, an indication of how healthy the fetus was, and how many children they subsequently had that in turn reached puberty. They found 16 years where the percentage of male infants surviving plunged, with one in the late 18th century dropping to 79 males for every 100 females.

[Gertie]

Eduk

Also she has two children. Not at least two children.

Just checking



Killosopher

Gertie, the birth order matters because had the first been a boy it would have allowed me to eliminate 2 of the sample spaces {GG} and {GB}. But since the birth order is not known I can only eliminate 1 sample space {GG} as both can't be girls since one is already a boy, which leaves me with 3 possible out comes {BG}, {BB} and {GB} making the probability 1/3 = 0.33... = 33.3%

I can follow how that makes sense, if you do your first calculation from a position of ignorance (as if it's before the woman had kids), then adjust it in the light of new info (one is a boy).

But if you start your calculation from the currently known position that a woman has 2 kids, one is a boy and the other unknown, and ask what are the odds of the unknown being a boy, you get 50/50. Cos every individual coin toss scenario has a 50/50 chance, regardless of what went before or after.
No?

Why the answers would be different I don't know, but I'm not a mathy type.

Then again it could be like the Monty Hall game show door paradox thingy, which I could never get my head round, so I'm not sure. https://betterexplained.com/articles/un ... l-problem/

[Fooloso4]

Gertie:

Then again it could be like the Monty Hall game show door paradox thingy, which I could never get my head round, so I'm not sure.

I’ve just been playing around with the Monty Hall game. I think I understand the math and the simulations support the conclusion but I am still reluctant to accept the claim that it is better to switch.