Multilevel Probability Question

Yes, very rare

For a simple random sample without replacement of 120 teachers from the
2000,
the probability that every teacher comes from a different school is:

#  probs[k] is the probability that the kth drawn teacher is not from any
of the previously drawn teacher's schools
# and equals 1 - the probability that the teacher drawn IS from one of the
previously drawn teacher's schools.
# After drawing one teacher from a school there are 4 left from that
schoold, so after drawing k-1 teachers there are 4*(k-1) teachers
# out of 2000-k-1 teachers that could be drawn

probs<-1 - 4*(0:199)/(2000-0:199)

# The probability that all of the 120 teachers are from different schools
is the product

prod(probs[1:120])

[1] 8.258304e-08
# picking 27 teachers from different schools is reasonably likely

prod(probs[1:27])

[1] 0.4861363

Rob



                                                                           
             Chuck Cleland                                                 
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                                                                   Subject 
                                       Re: [R-sig-ME] Multilevel           
             06/12/2007 08:51          Probability Question                
             AM

If you aren't stratifying by school, then isn't it just the
probability of drawing 120 teachers out of 2000?
-Hank

Creating the population described and taking a simple random sample of
120 teachers, it seems it would be very rare for no two teachers to come
from the same school:

pop <- data.frame(TEACHER = 1:2000, SCHOOL = rep(1:400, 5))

table(replicate(1000000, any(table(pop[sample(2000, size = 120,
replace=FALSE),]$SCHOOL) > 1)))

   TRUE
1000000

  There are obviously some samples where every teacher would come from a
different school, but I'm not sure how you might find the specific
probability.

hope this helps,

Chuck

On Jun 11, 2007, at 4:37 PM, Roberts, J. Kyle wrote:

Dear Friends,

I sent the following to the multilevel listserv, but thought that
some of you might actually have code to compute this in R.  If you
do, please let me know.

I have an odd question. I am trying to compute the probability of
drawing 120 teachers from a sample of 2000 teachers in 400 schools
where no two teachers work at the same school. Assume that there
are 5 teachers at each school, evenly spread. I was trying to do
this with an "n choose k" type
situation, but I can't figure out how to include the part about
only 5 teachers can be at any given school. Any ideas? I am not
stratifying on school, just sampling teachers.

Thanks,

Kyle


***************************************
J. Kyle Roberts, Ph.D.
Baylor College of Medicine
Center for Educational Outreach
One Baylor Plaza, MS:  BCM411
Houston, TX   77030-3411
713-798-6672 - 713-798-8201 Fax
jkrobert at bcm.edu
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