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Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is greater than their product? Express your answer as a common fraction.

 May 14, 2024
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There are 5×5=25 equally likely outcomes when two numbers are selected from 1 to 5. We count the number of outcomes which satisfy the condition that the sum is greater than the product.

 

The only way the sum can be less than or equal to the product is if one of the numbers is 1. So, the favorable cases are when we pick two numbers and neither of them is 1. There are (24​)=6 ways to pick two numbers from 4, and so there are 6 favorable outcomes. The probability is then $ \frac{6}{25} = \boxed{\frac{6}{25}}$.

 May 26, 2024

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