Question 378733
For the GCF to equal 2, n must divide 2, and n must not divide 3 nor 5. First, note that there is a 1/2 probability of n being even. Now we will consider the set of even integers 2, 4, 6, 8, ... and compute the probability that a chosen number *will* be divisible by 3 or 5 (then subtract from 1).

By inclusion-exclusion principle, the probability that a number n is divisible by 3 or 5 is 1/3 + 1/5 - 1/15, or 7/15. 1 - 7/15 = 8/15, times 1/2 gives us a probability of 4/15.


For the GCF of n and 60 to not equal 1, n must divide any of 2, 3, or 5. This requires some inclusion-exclusion bashing.

We can compute this by P(2|n) + P(3|n) + P(5|n) - P(2,3|n) - P(2,5|n) - P(3,5|n) + P(2,3,5|n)

= 1/2 + 1/3 + 1/5 - 1/6 - 1/10 - 1/15 + 1/60

= 43/60

I hope that's the right answer :)