Let's say that the universal set U has p items inside it where p is some positive integer
Define positive integers m and n such that
Set A has n items inside it
while set B has m items inside it
Every item inside set B is also found in set A. This makes set B a subset of set A.
The probability of selecting an item from set A is since we have n items from set A and p items from set U.
The probability of selecting an item from set B is since we have m items from set B and p items from set U.
Because , this means that .
We can divide both sides of the first inequality () by p to get the second inquality ().
The inequality sign will not flip because p is a positive number.
So this proves that P(B) <= P(A) is true, since,
P(A) = n/p
P(B) = m/p
(