Gender Island

[md65536]

If you make T,T,T, (after about 30 tosses, I got it), I can bet a lot that the next toss will be H. (in my small experiment I won my bet)

Or in the unlikely event you get T,T,T,T,T,T,T,T,T,T,T,T,T,T,T,T,T,T,T, I can bet Greece's debt that the next toss will be H (assuming this is still just a random case).

 

See: http://en.wikipedia.org/wiki/Exponential_distribution#Memorylessness

 

Essentially, an exponential distribution ... uh... i think you would say that it "looks the same" at any point as it did in the beginning. So the chances that next three tosses will be TTT is the same after 1000 tosses as it is after 1 toss.

 

The history (memory?) of what has just happened has a great influence on the total (historical) outcome of the random process, but it doesn't have an influence on what happens next. So, while there's a 1/1024 chance of getting 10 tails in a row, if you've already got 9 tails in a row then the probability of getting a tail next is still 0.5... Ie. the probability of getting 10 T in a row, given the case that you've already got 9 in a row, is now 50%. The probability of the next 10 tosses being T is still 1/1024. (So at that point there's 1/1024 chance of getting 19 T in a row! That is... in the on average 1/512 cases of getting 9 T in a row, there's a 1/1024 chance of getting another 10 in a row, ie. overall 19 T in a row should happen on average 1 in every 2^9*2^10 = 2^19 (half million) cases).

 

 

Your bet would be bad I think, because you're letting the slim probability of what has already happened affect your perceived chances of future events. But once something has happened, no matter how improbable, the probability of its occurrence changes to 1 given new information that it has indeed occurred!

 

I might bet the debt of Greece that you wouldn't get 20 T in a row in one chance, starting now. But I wouldn't make that bet after you'd already got 19 in a row!

 

 

 

[robheus]

A simple approach is this: each couple can have at most 1 girl (so, 0 or 1), but there can be any number of boys, because they keep on trying untill they have their first girl and then stop. Which means that boys outnumber girls.

[imatfaal]

A simple approach is this: each couple can have at most 1 girl (so, 0 or 1), but there can be any number of boys, because they keep on trying untill they have their first girl and then stop. Which means that boys outnumber girls.

 

Simple - but on the whole wrong. John Cuthber's explanation is my favourite in the thread. I must admit I went the long way around as per the original article/answer

[md65536]

Simple - but on the whole wrong. John Cuthber's explanation is my favourite in the thread.

 

I agree. The expected proportion of girls to total children on the island is

0.5 --- as long as nothing biases the gender of any given birth, and given no other specific information about anything, like whether the process has ended or not --- since we have that the probability of each independent birth being a girl is 0.5.

 

 

 

Assuming that answer is accepted, is my original "erroneous" question essentially the same question?

 

To repeat: Are you more likely to find that

a) the number of girls is greater than or equal to the number of boys, or that

b) the number of boys is greater than or equal to the number of girls, or that

c) neither case is more likely than the other?

You are given no further information (how many couples there are, how long this has been going on, how many children a couple can realistically have, etc).

Edited June 22, 2012 by md65536

[ShadowAngel7]

Assuming no realism-based limits on the number of boys each couple can have, I'd have said greater odds of more boys than girls...there can be anywhere from 0 to infinity boys, but only as many girls as there are couples. The number of ways for there to be more boys than girls is therefore greater than the number of ways for there to be more girls than boys. But that ignores the decreasing likelihood of increasing successive male births. Taking that into account, expected balance is almost exactly equal.

[D H]

John Cuthber's answer is correct, with two assumptions:

- That girls and boys are born in equal proportions,

- That death rates for males and females are equal, at least up through child bearing age.

 

Neither assumption is quite valid. Males slightly outnumber females at birth, about 105:100. Death rates for males also slightly outnumber that for females at all ages. The sex ratio becomes extremely close to 1:1 by the time those babies in a 105:100 male:female ratio grow to child-rearing age. This 1:1 ratio at child rearing age is not a fluke. It's true for many species. It's called Fisher's principle.

 

However, this is a detail that is intentionally being ignored in this problem. The assumed ratio is 1:1. An after-the-fact action such as stopping at the first boy won't change anything. Suppose the question was modified so that a couple stops having babies after having the fifth girl or after having the first boy. Naively this would seem to bias things strongly in favor of girls. It doesn't. These after-the-fact actions are closing the barn door after that 50/50 cow has already escape.

 

The only thing that can change the sex ratio is to change that 1:1 ratio (more realistically, 105:100 ratio) at birth. Pregnant women might abort fetuses in a gender-based manner, or newborns might die during or right after childbirth (and not count) in a gender-biased manner. Both apparently are happening in China as an unintended consequence of the one family, one child rule and societal biases that favor boys over girls.

[md65536]

I'd have said greater odds of more boys than girls...there can be anywhere from 0 to infinity boys, but only as many girls as there are couples. The number of ways for there to be more boys than girls is therefore greater than the number of ways for there to be more girls than boys.

I think that's an important factor (for the second "erroneous" question only) but I think that it actually works counter to your intuition.

 

 

 

 

However, this is a detail that is intentionally being ignored in this problem. The assumed ratio is 1:1. An after-the-fact action such as stopping at the first boy won't change anything. Suppose the question was modified so that a couple stops having babies after having the fifth girl or after having the first boy. Naively this would seem to bias things strongly in favor of girls. It doesn't. These after-the-fact actions are closing the barn door after that 50/50 cow has already escape.

 

+1 insightful.

 

However, in some cases "after-the-fact, closed barn door" information can affect the probabilities.

For example, if you're told that "The process has been completed successfully and no one is getting pregnant any more" then you know that the last child born was a girl, and that information alone can skew the distribution (by an unknown degree, without knowing some other numbers).

 

 

 

Yes, the caveat of "assume an absence of any factors that unnecessarily complicate the puzzle" was meant to intentionally avoid the issue of biases and imply a 1:1 ratio without specifically saying it an giving the answer away.

 

 

[John Cuthber]

John Cuthber's answer is correct, with two assumptions:

- That girls and boys are born in equal proportions,

- That death rates for males and females are equal, at least up through child bearing age.

 

Neither assumption is quite valid. Males slightly outnumber females at birth, about 105:100. Death rates for males also slightly outnumber that for females at all ages. The sex ratio becomes extremely close to 1:1 by the time those babies in a 105:100 male:female ratio grow to child-rearing age. This 1:1 ratio at child rearing age is not a fluke. It's true for many species. It's called Fisher's principle.

 

However, this is a detail that is intentionally being ignored in this problem. The assumed ratio is 1:1. An after-the-fact action such as stopping at the first boy won't change anything. Suppose the question was modified so that a couple stops having babies after having the fifth girl or after having the first boy. Naively this would seem to bias things strongly in favor of girls. It doesn't. These after-the-fact actions are closing the barn door after that 50/50 cow has already escape.

 

The only thing that can change the sex ratio is to change that 1:1 ratio (more realistically, 105:100 ratio) at birth. Pregnant women might abort fetuses in a gender-based manner, or newborns might die during or right after childbirth (and not count) in a gender-biased manner. Both apparently are happening in China as an unintended consequence of the one family, one child rule and societal biases that favor boys over girls.

Good point about the possible difference in death rates, but I did mention that birth rates are not exactly 50:50 and that this sort of thing only really works in maths problems.

[D H]

However, in some cases "after-the-fact, closed barn door" information can affect the probabilities.

For example, if you're told that "The process has been completed successfully and no one is getting pregnant any more" then you know that the last child born was a girl, and that information alone can skew the distribution (by an unknown degree, without knowing some other numbers).

We're ruling out gender-biased abortions, gender-biased slips of the knife during birth, right? (These can and do bias the sex ratio.) Assuming that is the case, absolutely nothing done after the fact can change the sex ratio. Some scenarios:

- Parents just have babies, stop whenever they want (or biology says "stop"!): 50% boys, 50% girls.

- Parents must stop at the first boy, but must keep on having babies, maybe forever, until you have a boy: Still 50/50.

- Parents must stop at either the first boy or at the nth girl: Still 50/50, no matter what value one chooses for n.

- Parents can stop whenever they want except they must stop at the first boy: Still 50/50.

 

 

You want logic? The logic is pretty simple. The cow remains out of the barn if one closes the barn door after the cow gets out.

 

You want math? It's simple probability theory. No matter how one sets up the rules of the after the fact game, the sex ratio will remain 50/50.

 

You want a simulation? It's going to agree with the logic and the math.

Edited June 22, 2012 by D H

[md65536]

You want logic? The logic is pretty simple. The cow remains out of the barn if one closes the barn door after the cow gets out.

 

You want math? It's simple probability theory. No matter how one sets up the rules of the after the fact game, the sex ratio will remain 50/50.

 

You want a simulation? It's going to agree with the logic and the math.

 

Yes, you're right, but if you're given information about the cow that got out, that "new information" can have an influence on what you're describing.

 

I've done a simulation to confirm.

 

See also this very related (but much more complicated) problem: "I have two children.

One is a boy born on Tuesday. What is the probability I have two boys?"

http://www.physicsfo...ad.php?t=419102

 

 

 

 

Yes, you're right, that if you're told that a family will stop trying after N boys, that doesn't change anything.

If you're told that some family stopped trying after having N boys, that is new information that does have an influence.

If you're told that some family would stop trying after N boys but they didn't have to, that's new information.

 

As well... Suppose you're told that any family would stop after having N boys. If you are shown an example family that stopped, that is new information (it excludes all possible cases where all families had a girl) --- AND it is different information than being shown a randomly picked family and finding out that they stopped after having N boys. (This I think is the main point with the "I have two children" problem. Nah that's not the point, but whatever.)

 

 

PS. I'm going to add a new puzzle or two to discuss this point...

 

Edit: So, while writing out the "cow island" puzzles I figured something out:

 

 

 

If you say "every child born is equally likely to be a girl as to be a boy" and you set that in stone, then nothing will change that.

 

If you say "I have two children, each equally likely to be a girl or boy. The first is a boy." The second statement changes the information given in the first. Depending on wording or interpretation, you may say it contradicts the first statement.

 

The thing is though, we're assuming that all the children born on the island are each actually either a girl or a boy, and not some probabilistic superposition of the two possibilities. So I think that we must interpret new information, such as "(In some particular case) the last one born was a girl" as clarifying the possibility of the situation, not changing it.

 

Or in other words, if we specify the case (as with my example: the case where the process has ended successfully with every family having a girl) then the previous information "every birth is equally likely to be a boy or girl" should be considered to be incomplete information, not a fact set in stone.

 

 

 

Edited June 22, 2012 by md65536

[D H]

See also this very related (but much more complicated) problem: "I have two children.

One is a boy born on Tuesday. What is the probability I have two boys?"

http://www.physicsfo...ad.php?t=419102

The answer is zero in the case of your island with a one family, one son rule. There are no families with two boys.

 

 

Getting back to your island, you can't change the gender ratio without culling, and we've ruled that out.

[md65536]

The answer is zero in the case of your island with a one family, one son rule. There are no families with two boys.

 

 

Getting back to your island, you can't change the gender ratio without culling, and we've ruled that out.

 

You're mixing up both the puzzles, and the rules of the puzzle. The latest linked puzzle was meant to be treated separately. On the island, there may be families with two boys, but none with two girls. In the "google quiz" it's reversed.

 

 

Back to the island: You can change the measured gender ratio by manipulating when you decide to measure it! The expected number of girls to boys will remain 1:1 but the actual number of girls minus boys will be a random walk. If you say "I will only measure the gender ratio when there happen to be more girls than boys" then obviously you're going to get (quite) skewed results. Likewise, if you say "I will only measure the gender ratio after the birth of a girl" then you have a slighter and more subtle skew to the result.

 

In your examples, you don't explicitly state any bias, but others have (eg. the assumption that the process completes successfully) and it's important to either carefully avoid the biases or acknowledge their influence.