Timeline for Google question: In a country in which people only want boys
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 11, 2010 at 17:42 | comment | added | T.. | @Douglas: what followed "obviously" was the observation that it is not possible for stopping to enforce a surplus of boys in the population, although it can certainly enforce this, eventually, within each family. If you define "population" (i.e., the set of coin flips over which G/(G+B) is calculated) in terms of the stopping rule, as you did with your model with completed families only, then of course what is true for the stopping rule can be true of the population, but it would be hard to argue that the problem corresponds to any such model. | |
| Jul 10, 2010 at 4:12 | comment | added | Douglas Zare | @T: What follows "obviously" in your response is false. With probability 1, the families can stop with a surplus of boys. The optional stopping theorem has a finiteness assumption which is necessary and which is violated in this case, so the conclusion does not hold. It is possible to choose a stopping rule on a stream of fair coin flips so that you stop with probability 1, and when you stop, there are more heads than tails. The expected number of tosses for this stopping rule must be infinite. Again, the proportion of girls is not a martingale, so you can't expect the OST to apply anyway. | |
| Jul 10, 2010 at 4:04 | comment | added | Douglas Zare | @Daniel Asimov: 1 and 2) In that case I disagree with the statement. The proportion of girls in the population is not a martingale, and so it should not be a surprise that there are stopping rules which change the expected value. 4) A stopping rule suggested by your modification is that each family stops with at least one child and at least as many boys as girls. Another is that each family stops with 1 more boy than girl. With a finite number of families, the expected proportion of boys in the generation is not 50%. E[B] would not exist, but E[B/(B+G)] would. | |
| Jul 9, 2010 at 16:46 | comment | added | T.. | Douglas, if every family uses a stopping rule that enforces B > G (with probability 1, such as "reproduce until B > G") this obviously does not and cannot enforce a surplus of boys in the population. It is especially clear if you replace families/children by the isomorphic setup of gamblers/cointosses. If every gambler follows a "play until ahead" strategy that in no way implies a loss for a casino offering fair games. It does alter the allocation of tosses (children) to gamblers (families) but for the casino (population) the allocation is irrelevant. | |
| Jul 8, 2010 at 23:43 | comment | added | Daniel Asimov | 1. The words "a social convention cannot override biology" (not mine) mean just that the ultimate proportions of boys and girls are the same as the proportions in which boys and girls are born. 2. The stopping rule (in question) is a red herring because any stopping rule of the form "Stop as soon as a certain consecutive string of B's and G's occurs" will result in the same ratio of 1:1 (or more generally p:q) as the probabilities of B vs. G (or H vs. T) are in. 3. You are right that the 2:1 stopping rule is not almost certain to occur. Oops. 4. What modification are you thinking of? | |
| Jul 8, 2010 at 23:27 | history | edited | Daniel Asimov | CC BY-SA 2.5 |
Removed false statement.
|
| Jul 8, 2010 at 19:37 | comment | added | Douglas Zare | If you modify that example, then each family can stop with at least as many boys as girls with probability 1. That means with probability 1, the country's population will have at least as many boys as girls, and some populations will have more boys than girls. | |
| Jul 8, 2010 at 19:33 | comment | added | Douglas Zare | The stopping rule matters because the ratio G/(G+B) for the population becomes a biased estimator of the probability that each child is a girl. Just as it is biased for 1 family, it is biased for 2, 10, or any finite number of families. Each boy or girl is weighted by 1/(population size), and girls tend to belong to larger populations. I'm not sure what you mean by "a social convention cannot override biology" or "clearly a red herring." If a family stops when there are at least twice as many girls as boys, then with positive probability (I believe 1/2 phi^-1) the family will not stop. | |
| Jul 8, 2010 at 19:06 | history | edited | Daniel Asimov | CC BY-SA 2.5 |
Mentioned stopping condition for which the previous reasoning fails
|
| Jul 8, 2010 at 15:37 | history | answered | Daniel Asimov | CC BY-SA 2.5 |