Need some math help

[montanafan]

I need some help in a debate I'm having with someone. I say that if you have a city with 10 times the population of another, it should be much easier to find 12 good athletes in the larger city. Can anyone here give me some help with the probabilities?

Case 1:
Town A - population 4,500 and Town B - population 47,427

Case 2:
Town A - population 4,500 and Town B - population 200,073

How would you show that it is more likely that you could find 12 good athletes in the larger town for each case?

Edit: Typo

 

[summit]

in theory this would be true, however if its real life this is way off. china 1 billion people, india 1 billion people, america 250 million people, we got some athletes!

you just do it there's 1 good athlete out of every 900 ... Case 1: 5 good athletes for town A ... 52 good athletes

Case 2: 5 good athletes town A and 222 good athletes

this is just the probability of finding good athletes assuming conditions are equal, nutrition, climate, genetic variation. however if this is applied to real life all this is tossed out the window. there are too many variations of how schools like Long Beach Poly have many bluechip football recruits compared to citys of the same size. or the usa to china/india aspect. it depends on money too.

 

[montanafan]

Thanks summit, I should have thought of just substituting the numbers.

I know what you mean about the difference between reality and statistics, but in this particular case the reality favors my side of the debate as well. The cities are in the same state, so the demographics are closer than between different countries/cultures, and money and opportunity would be on the side of the larger population area too. Thanks again.

 

[DrPizza]

Originally posted by: Summit
in theory this would be true, however if its real life this is way off. china 1 billion people, india 1 billion people, america 250 million people, we got some athletes!

you just do it there's 1 good athlete out of every 900 ... Case 1: 5 good athletes for town A ... 52 good athletes

Case 2: 5 good athletes town A and 222 good athletes

this is just the probability of finding good athletes assuming conditions are equal, nutrition, climate, genetic variation. however if this is applied to real life all this is tossed out the window. there are too many variations of how schools like Long Beach Poly have many bluechip football recruits compared to citys of the same size. or the usa to china/india aspect. it depends on money too.

People are born with natural ability, but the environment shapes that ability. Most people never perform up to their potential. There are plenty of examples of small towns, for instance, that have outstanding football teams year after year after year and beat up on the much larger schools that they play. It's not something in the water, but rather that they have been developed much better.

 

[summit]

Originally posted by: montanafan
Thanks summit, I should have thought of just substituting the numbers.

I know what you mean about the difference between reality and statistics, but in this particular case the reality favors my side of the debate as well. The cities are in the same state, so the demographics are closer than between different countries/cultures, and money and opportunity would be on the side of the larger population area too. Thanks again.

what inner city school vs. huge suberban school? g

 

[montanafan]

Summit, it's a small rural school and a small urban school. One has a student body of 210 and the other 336, but the 336 has a much larger population area to draw eligible students from.

 

[Parasitic]

There's a couple of ways to look at this problem. This may be the easiest and quickest.

Suppose that each individual has an equally likely chance to be a good athlete, with said probably p. Assuming that conditions at both populations are the same such that p is constant in both conditions. Then each individual in the population can be modeled as a Bernouli random variable, and the population as a whole can be modeled as a random variable with binomial distribution bino(n,p). The expectation of this random variable is E(X) = np. Now for simplicity's sake let's say town A produces exactly 12 good atheletes, then E(A) = 12 = np, so the probability of an individual being a good athlete is ~ p=12/4500 or .26%; then in town B, the expected number of good athletes will be E(B) = 47427*p = 126. Do the same for your second case.

Alternatively, you can use Student's t-dstribution to set up a hypothesis testing where H0: mean(A)-mean(B) = 0 and HA: mean(A)-mean(B) != 0. Then choose an type I error to be .05 and determine the number needed to shift the mean.

 

[montanafan]

Originally posted by: Parasitic
There's a couple of ways to look at this problem. This may be the easiest and quickest.

Suppose that each individual has an equally likely chance to be a good athlete, with said probably p. Assuming that conditions at both populations are the same such that p is constant in both conditions. Then each individual in the population can be modeled as a Bernouli random variable, and the population as a whole can be modeled as a random variable with binomial distribution bino(n,p). The expectation of this random variable is E(X) = np. Now for simplicity's sake let's say town A produces exactly 12 good atheletes, then E(A) = 12 = np, so the probability of an individual being a good athlete is ~ p=12/4500 or .26%; then in town B, the expected number of good athletes will be E(B) = 47427*p = 126. Do the same for your second case.

Alternatively, you can use Student's t-dstribution to set up a hypothesis testing where H0: mean(A)-mean(B) = 0 and HA: mean(A)-mean(B) != 0. Then choose an type I error to be .05 and determine the number needed to shift the mean.

Well...when you put it like that, sure. I'd have no idea how to explain that, but I do appreciate the help.

 

[summit]

doesn't really matter because your student population is so small. neither schools recruit so your student population is your constant, the variable is how many athletes there are. since the difference is only 100, it doesn't really matter.

 

[montanafan]

I see what you're saying, but let's say that each school can pick and choose who they want to enroll and both are looking for athletes. Since the one school can only draw students from a population of 4,500 and the other can enroll anyone from a population of over 200,000, if both are looking for athletes, wouldn't the one with the larger enrollment area have an advantage in the numbers they could find?

 

[Parasitic]

Originally posted by: montanafan

Originally posted by: Parasitic....

Well...when you put it like that, sure. I'd have no idea how to explain that, but I do appreciate the help.

Well you wanted an explanation by probability...

 

[montanafan]

That's true. Perhaps I should try another tactic in my debate.

 

[summit]

Originally posted by: montanafan
I see what you're saying, but let's say that each school can pick and choose who they want to enroll and both are looking for athletes. Since the one school can only draw students from a population of 4,500 and the other can enroll anyone from a population of over 200,000, if both are looking for athletes, wouldn't the one with the larger enrollment area have an advantage in the numbers they could find?

how would this be possible, but if possible yes you could use it as an argument, however the counter argument will always be the smaller population is some how more entrained for athletics (nature v. nuture). but seriously how do schools pick from the population. unless something is shady/private school/magnet school.

 

[FleshLight]

Assuming the probability of getting an athlete is constant, then you can use the expected value to determine the expected # of atheletes.

expected value = (population) * (probability)

200,000p > 15,000p > 4,500p

 

[Safeway]

Originally posted by: FleshLight
Assuming the probability of getting an athlete is constant, then you can use the expected value to determine the expected # of atheletes.

expected value = (population) * (probability)

200,000p > 15,000p > 4,500p

I don't know if I would listen to a FleshLight. :evil: