What Field Studies Small Groups with a Characteristic?

Date: 6/10/96 at 17:29:34
From: Anonymous
Subject: What field studies small groups with a characteristic?

I'm interested in knowing whether there is a field that studies the 
following or if there is a solution to this question.  I have not been
able to find any useful information in a textbook.

1. Take for example that person A has blonde hair.
2. You know that this person is not famous.
3. You also know that (10) people know person A.
4. You would like to know that if (10) people know person A, how many 
   people in a given population also know someone like person A, 
   someone with blonde hair.
5. You really want to know how many people with blonde hair there are 
   in a given population given the answer to #4.

Thank you.

Date: 7/1/96 at 8:41:28
From: Doctor Moskowitz
Subject: Re: What field studies small groups with a characteristic?

Yes, there is certainly a branch of math that deals 
with these types of questions - it's Probability.  Any question of 
determining information about a large number of objects (in this 
case people) from information about a smaller sample is the province 
of  probability and statistics.

In the case of your problem, you need to be a little more specific 
about some things.  It makes a difference whether you ask if a person 
in the population "knows a person with blond hair" or ask if they know 
_exactly_one_  person with blond hair.  In the latter case, if 
everyone knows ten people, and exactly one has blond hair, then 10% of 
the population are blondes.  If the former is true, you'd need to find 
out exactly how many blondes each person knew to get a definite answer 
to your question.

However, this is a more special case than you'd usually expect, since 
we somehow knew what every single person knew, and further knew that 
every single person knew exactly ten people.  In the real world that's 
ridiculous, different people know different numbers of folks; even if 
nobody's "famous" there are still people who are known by more than 
others; you usually only have access to a certain small portion of the 
population.  And that's where a huge amount of mathematics comes in -- 
the fields of probability and stat. The answers given to problems with 
uncertainties in them are themselves uncertain -- that is, there's a 
"most likely" answer, along with another number indicating how much 
that number might be off from reality.

To get further info about statistics, you might want to check out
 http://olam.ed.asu.edu/~glass/502/home.html   
- it's an entire statistics course on the Web!

-Doctor Moskowitz,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/