Let us say that the probability (p) of picking one of the 200 courses is:
p = 1 / 200 = 0.005
Therefore using the binomial probability (P), we can solve for the probability that at least 2 professors pick the same course. However what we will do it the opposite, calculate for the P when 0 and 1 professor only pick that course then subtract the sum from 1.
The formula for P is:
P = [n! / r! (n – r)!] p^r * q^(n – r)
where, n is the total number of professors = 100, r is the number of professors who picked the same course = 1 or 0, p = 0.005, q = 1 – p = 0.995
when r = 0
P = [100! / 0! (100 – 0)!] (0.005)^0 * (0.995)^100
P = 0.606
when r = 1
P = [100! / 1! (100 – 1)!] (0.005)^1 * (0.995)^99
P = 0.304
Therefore total P is:
P = 0.606 + 0.304
P = 0.91
Hence the probability that 2 or more will pick the same course is:
1 – 0.91 = 0.09
= 9%
Therefore there is a 0.09 or 9% probability that at least 2 will pick the same course.