Martha Huntzicker
Problem: Two students are randomly chosen from a regular elementary school
class. They both have blue eyes. If the odds are exactly 50/50 that two randomly
chosen students in the class have blue eyes, how many students are in the
classroom?
Solution:
Let n= the total number of students.
Let b= the total number of blue eyed students.
The probability that two students with blue eyes are chosen at random
simultaneously and/or without substitution is as follows:
[(b/n)]*[(b-1)/(n-1)]=[b(b-1)]/[n(n-1)]
This probability is equal to 1/2 (the odds are exactly 50/50).
Therefore, to determine the number of students in the classroom, we must find
values of b and n that satisfy:
[b(b-1)]/[n(n-1)]=1/2
There are many pairs (b,n) that satisfy this expression:
(3,4)
(15,21)
(85,120)
.....
Intuitively, there are most likely 21 students in the classroom, fifteen with blue eyes. It is unlikely that a
classroom has four students (unless they are home schooled); and I sincerely hope there are no
classrooms with one hundred twenty students at the elementary school level.