Two problems that were on my job application

After a night of wild revelry, a group of local intoxicated hoodlums stumble into a children's playground. One of these hoodlums, Phil, climbs to the middle of a spin-around carousel and his friends push the carousel so it rotates once every five seconds. Phil, who is trying not to get sick, holds a flashlight motionless in his hand.

There is a straight path running by the playground that, at its closest point, is 30 meters from the middle of the carousel. Unknown to Phil, there are two cops facing him on the path, shocked at the spectacle. One of them is standing on the path at the point closest to the carousel, while the other is standing 60 meters down the path. At approximately what speed (in meters per second) does the spot illuminated by the flashlight traverse each of the cops' bodies? State speed at the closest cop first.


A rustic village contains one million married couples and no children. Each couple has exactly one child per year. Each couple wants a girl, but also wants to minimize the number of children they have, so they will continue to have children until they have their first girl. Assume that children are equally likely to be born male or female. Let p(t) be the percentage of children that are female at the end of year t. What is p(t)? "Can't tell" is a potential answer if you don't have sufficient information.


Let's see how fast you guys can answer these.

#1

Angular velocity is 2 pi/5 radians/sec. 

First cop is 30 meters away.

Second cop is sqrt(30^2+60^2) = 67.08 meters away.

Velocity of spot at first cop = (2 pi * 30 meters/radian) * (2 pi / 5 radians / sec) = 237 meters / sec

Velocity of spot at second cop = 530 meters/sec

#2

Something to do with the central limit theorem and a truncated geometric distribution, but I hate stats and am too lazy to do it right now. :P

production rate of village = 1 child per couple per year
hence there are a great number of couples there will be equivalent children
year,population without girl,rate
1,500T,0.5
2,250T,0.75
3,125T,0.875875
approx It is p(t)=p=0.5
Large number e function approximation: Couples without girls = 1*exp(-0.5 *t)
Approximation
Resulsts may vary.

For the second one:

We only need to consider one couple since they're all independent.
For each couple, the chance of a girl being born at the Tth year is 2^-T.

After T years, the probability distribution for the number of children per couple is given by
q(k) = 2^-k if  k < T
q(T) = 2^-(T-1)

So the mean number of children per couple after T years is
(1 * 1/2 + 2* 1/4... +  (T-1) 1/2^-(T-1)) + T*2^-(T-1)
(find a nicer expression yourself).

The average number of girls among these is 1-2^-T, so divide this quantity by that series and there's your answer.

typo*
(1 * 1/2 + 2* 1/4... +  (T-1)2^-(T-1)) + T*2^-(T-1)

>A rustic village contains one million married couples and no children.

That's pretty big for a village.

Straightaway we can see that as T tends to infinity, p(T) tends to

1/ 1/(1-1/2)^2 = 1/4

Oops.
The mean number of children as T tends to infinity is
x+2x^2 + 3x^3... evaluated at x = 1/2
= x(1+2x+3x^2...) = x d/dx (x+x^2+ x^3) = x d/dx x/(1-x)
 = x d/dx (1/(1-x) - 1)
= x/(1-x)^2 = 2

So p(T) tends to 1/2 not 1/4.