Restricted choice?

[blackshoe]

The OP's daughter said "I have three children..." IOW, she has three children. Nothing to do with how many her father has. "And one of them is a boy." As for computing probabilities, the simple question doesn't give enough data. You're either going to have to get more data, or make some assumptions. In the latter case, it would be useful to know what those assumptions are. B)

[gwnn]

http://en.wikipedia.org/wiki/Sex_ratio

 

says there are 105 to 107 boys for 100 girls born

[kenberg]

It isn't clear to me that the "natural" interpretation of the problem is "we choose a family at random from all the 3-children families which have at least one boy" rather than "we choose a random child from all the families which have 3 children and it turns out to be a boy"

Upon reflection, I guess I agree. The statement focuses on the parent having 3 children one of which is a boy, rather than on the boy, so I think that indicates a randomly chosen family (or parent, which is the same thing) but I concede that it is not exactly an unchallengeable interpretation.

 

 

As to zero:

I was of course joking a bit about the probability being zero because his daughter asked him, and stretching a bit to make the father the referent of the pronoun I. There are no quotes around the statement so I figured I could argue the point, but w/o serious intent.

 

 

Moral of the story: Probability problems very often come down to describing precisely what is being analyzed. Deterministic problems often (but not always) are forgiving of vague formulations. Probability problems are ruthless.

[EricK]

Moral of the story: Probability problems very often come down to describing precisely what is being analyzed. Deterministic problems often (but not always) are forgiving of vague formulations. Probability problems are ruthless.

This was really my point. The fact that they gave the answer of 1/7 means they had a certain interpretation in mind. However the fact that they didn't make all the assumptions clear in the problem means that there is a fair chance they don't understand what assumptions underlie their answer and why they are important.

 

And that is worrying if this question came from someone trying to teach probability.

[Fluffy]

No Gonzalo, that's different. If you only examined one of the kids and that kid turned our to be a boy, the there are only four combinations for the remaining two kids.

I agree but I fail to see where is the other appraoch failing.

[gwnn]

I think these probability problems are a bit... random (pun intended).

 

Like the non black non raven paradox, it depends if you were looking for a random non black thing that turned out to be a non raven OR a random non raven that turned out not to be black. These delimitations sound a little ridiculous to me. Well maybe the paradox was also a little ridiculous.

[kenberg]

Moral of the story: Probability problems very often come down to describing precisely what is being analyzed. Deterministic problems often (but not always) are forgiving of vague formulations. Probability problems are ruthless.

This was really my point. The fact that they gave the answer of 1/7 means they had a certain interpretation in mind. However the fact that they didn't make all the assumptions clear in the problem means that there is a fair chance they don't understand what assumptions underlie their answer and why they are important.

 

And that is worrying if this question came from someone trying to teach probability.

It may be a thread hijack to go far into this but I see this as an instance of a fairly major problem, especially in regard to standardized tests. Far too often, the questions are vaguely phrased but students are expected to interpret them in a specific manner. If a student wants to do well, it is best that he not think deeply about the question. Rewarding superficial thought and punishing reflective questioning is presumably not the intent of the exams but I think that it often comes to that. Back in my school days, my fundamental rule for multiple choice exams was to never give any serious thought to any question. This strategy worked well.

[1eyedjack]

I find it interesting that there are more boys born than girls. Does this mean that there are more boys in total than girls? Not necessarily.

 

Parents with three daughters try once more

and then it's fifty-fifty they'll have four.

But those with a son or sons will let things be.

Hence all these surplus women, QED.

 

Actually someone told me that the relative birthrate depends on a number of external factors such as wars, diet, diets resulting from wars etc. I don't know if that is apocryphal

[helene_t]

I find it interesting that there are more boys born than girls. Does this mean that there are more boys in total than girls? Not necessarily.

 

Parents with three daughters try once more

and then it's fifty-fifty they'll have four.

But those with a son or sons will let things be.

Hence all these surplus women, QED.

I don't think that influences the sex ratio much.

 

By far the most important reason for the lack of male surplus is that boys are more likely to die than girls are.

[helene_t]

No Gonzalo, that's different. If you only examined one of the kids and that kid turned our to be a boy, the there are only four combinations for the remaining two kids.

I agree but I fail to see where is the other appraoch failing.

Suppose that BGG means that the first kid whose sex you determine is a boy while the too later were girls; if it was the last kid to have its sex to determined that was a boy, you would write GGB.

 

Then the event that one of the three children is known to be a boy while the two others are yet to be decided, leaves only four possible combinations:

 

BBB

BBG

BGB

BGG

 

So you have to distinguish between these two scenarios:

 

1) Only one child had its sex determined and it turned out to be a boy. The sex of the other two children have not been determined. In that case, the chance that all three are boys is 1/4.

 

2) The sex of all three children is known and at least one of them is a boy. In that case, the answer is 1/7.

 

If the information given is simply "at least one of the children is a boy" then there is no answer. You have to be more specific about what it is that you know.

[Fluffy]

you are repeating the same, already stated before. I know that's right, but I expected my approeach to add to 3/12 to get 1/4 as suposed, that 3/13 is making me crazy, as I don't know where the problem is.

[kenberg]

Referring to Jack's jingle. Posts are coming fast and furious. I thought this would be directly following Jack's.

 

As you may be aware, the jingle above draws the wrong conclusion. Put aside for the moment the fact that parents with 3 children of the same sex might in fact have a genetic predisposition for having children of that sex. Assume, as the jingle says, it is 50-50. Well, 50-50 is 50-50. If each pregnancy has a 50-50 chance of producing a girl then trying again with three daughters but letting it be with at least one boy will still mean that the pregnancy has a 50-50 chance of introducing another girl. Such a policy would influence the distribution of families with various configurations. For example, if followed strictly then there would be no families with three girls and two boys (unless the boys were twins) because by the time they had four children they would have had at least one boy and they would let it be. But influencing family configurations is not the same as influencing the overall percentage of girls.

 

Now if we introduce the possibility that having three girls is an indicator that the next pregnancy has a better than 50-50 chance of being a girl then yes, following this policy could increase the percentage of female births.

 

So: If you believe that families often keep at it until they have at least one boy then the fact that boys are more than 50% of the births would be some (weak) evidence against the idea that having three girls indicates a genetic predisposition for giving birth to girls.

[helene_t]

you are repeating the same, already stated before. I know that's right, but I expected my approeach to add to 3/12 to get 1/4 as suposed, that 3/13 is making me crazy, as I don't know where the problem is.

Your numbers add up to 12 so it would be 3/12 which is correct.

[EricK]

Boys are more prone to die than girls for a number of reasons. eg

 

They are more likely to be involved in violent incidents.

They are more prone to various genetic disease (because they only have one X chromosome and where a girl might be able to compensate for a defective gene on one X chromosome with a normal gene on the other, a boy can't)

They are more prone to risk taking

They are genetically programmed to spend less resources on repairing the body and more resources on being vigorous in youth and middle age (essentially for the reasons above)

 

Because of this, although men outnumber women at birth, women end up outnumbering men in almost every age range.

 

Where a society has a surplus of men, it is always because of female infanticide (whether after birth, or via selective abortion)

[irdoz]

There are studies which show a very small but significant decline in the birth rate of boys including in the USA, Canada and the Netherlands. The journal article in the AMA which reviewed the international studies theorised that it was caused by 'sex hormone pollutants'.

 

Dioxin pollution causes a more significant drop in the male birth rate. This has been observed in one Russian city and in some parts of SE Asia.

 

There's a small community in Canada where the male birth rate has gone from 52% in 1993 to 33% some years later. It is located near a large industrial area and the exact reason is unknown.

[helene_t]

Yeah, there was also once talk about radiology nurses in Denmark having something like 90% girls. Presumably boys are more vulnerable to radiation because of the single X chromosome. Dunno if those studies have been reproduced.

[kenberg]

From the 2000 USA census

 

www.census.gov/prod/2005pubs/censr-20.pdf

 

 

Here are some figures. Going across the numbers are: Age group; # of men ; # of women; percentage of men in that age group

 

0-4 &&& 9,755,707 &&& 9,291,047 &&& 51.2%

 

30-34 &&& 10,219,811 &&& 10,145,302 &&& 50.2%

 

60-64 &&& 5,114,578 &&& 5,673,401 &&& 47.4%

 

85+ &&& 1,203,376 &&& 2,957,185 &&& 28.9%

[barmar]

From http://www.drdaveanddee.com/longevity.html

Life expectancy tables give average life expectancy depending on year of birth. For example, life expectancy for someone born 2009 is 80.0 for females and 74.88 for males, an age difference of about 5 years (Social Security Administration at www.ssa.gov, 2009).

 

However, life expectancy increases with age, and the male/female difference decreases. For example, someone who is 60 years old in 2009, a female's life expectancy is 83.34 and a male's is 80.01, a difference of about 3 years. But, someone who is 100 in 2009 has a life expectancy of 102.36 if female and 102.02 if male, a difference of 0.34.

I think the reduction in the male/female difference as you get older is because many of the factors that reduce male life expectancy apply mostly to younger men. Violence tends to be more extreme among teenagers and twenty-somethings. Congenital problems are more likely to kill children (put another way, if you make it past childhood, it probably isn't serious enough to kill you).

[helene_t]

But Barmar, even if male mortality is always higher than female mortality, for example by a factor (say) 1.1 regardless of age, you would observe that the female/male ratio gets higher the higher the age.

 

What you say is probably true, though, except that in some countries maternal death during labor causes mortality to be higher among females than males in a certain age range. And, as have been mentioned by others, in some countries baby girls are more likely to die than baby boys, due to neglect or outright murder.