hi, does anyone know the anwser for this ? thanks!A mythical city contains 100,000 couples but no children. Each family wishesto "continue the female line" but they do not wish to over-populate. So eachfamily has one baby per annum until the arrival of the first girl. Forexample, if (at some future date) a family has five children then it must beeither that they are all boys, and another child is planned, or that there arefour boys and one girl, and no more children are planned. Assume the childrenare equally likely to be born male or female. Let p(t) be the percentage ofchildren that are female at the end of year t. How is this percentage expectedto evolve through time? What is the average number of children per family?

50%constant thru time2

hi, can you explain why?

Since there is 50 % of having either a boy or a girl , taking into account the law of large numbers,you can see that every year about 50% of the births will be girls so the % of girls remains constantover t (at the first year you have 50% girls and 50% boys , the next year all those that had boyswill give birth once more, again 50 % of those will be girls and 50 % boys, and so on... so % of girlsremains constant).If you can't see right away why the average # of births per family will be 2, look at a random processB(t) of births where every year you have probablity 0.5 of having a boy or a girl. Define the stoppingtime t_s to be the time of having the first girl. You need to calculate E[Bt_s] :E[Bt_s] = sum_{i=1 to \infty} (1/2)^i * i = 2.There are a lot of stopping time problems in the brainteasers forum (try searching for 'coin' forexample).

Another way to get second answer is just to look at the explanation of the first answer. When couple has a kid, they have 50% chance to get another one, after that, they have 50% change to get another .... ie. the average number of kids per couple is 1 + 1/2 (1+ 1/2 (1+ 1/2 (1+ ... if this is to be convergent you get Sn = 1 + 1/2 Sn => Sn=2

QuoteOriginally posted by: mememehi, does anyone know the anwser for this ? thanks!A mythical city contains 100,000 couples but no children. Each family wishesto "continue the female line" but they do not wish to over-populate. So eachfamily has one baby per annum until the arrival of the first girl. Forexample, if (at some future date) a family has five children then it must beeither that they are all boys, and another child is planned, or that there arefour boys and one girl, and no more children are planned. Assume the childrenare equally likely to be born male or female. Let p(t) be the percentage ofchildren that are female at the end of year t. How is this percentage expectedto evolve through time? What is the average number of children per family?You need this book:Heard on the Street: Quantitative Questions from Wall Street Job Interviews by Timothy Falcon Crack I has a lot of brainteasers (including your one) with answers.

Some of these questions are set up to trick you into believing that you will be working on cool puzzles on the job. Don't be fooled by these questions -- they have very little to do with what you actually will be doing (that is, slinging reams of code at the whim of the traders and other bosses). Also, beware: if you appear too knowledgable solving these esoteric puzzles, you will be labelled and paid at the mathematician scale. Once you exhibit too much mathematician abilities, you are ruled out from ever being trader or portfolio manager material.