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counting and probability intermediate
Problem
Two distinct numbers are selected simultaneously and at random from the set . What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.
Solution
There are 10 pairs of integers that we can potentially select. The easiest way to do this is to simply write them all out: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), and (4,5). The 4 pairs with 1 as an element all obviously work, as does (2,4), but none of the others do. That means that 5 out of 10 pairs work which gives us a probability of .
Final answer
\frac{1}{2}