Question

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 less than their product? Express your answer as a common fraction.

Answer 1

Answer: P = 15/25

Explanation:

The set of numbers that we have here is:

{1, 2, 3, 4, 5}

We select independently two numbers of that set (so the numbers can be repeated.

We want to find the probability where the sum of the numbers is less than the product.

if one of the selected numbers is 1, then always the sum will be larger than the product, because:

1*1 = 1 and 1 + 1 = 2

1*2 = 2 and 1 + 2 = 3.

and so on.

if both numbers are 2, the sum is equal to the product:

2*2 = 4 = 2 + 2.

if else, the product will be larger than the product.

The first step now is to calculate the total number of possible combinations of 2 numbers:

For the first number, we have 5 options.

for the second number, we have 5 options.

The total number of combinations is equal to the product of the number of options in each case:

C = 5*5 = 25

Now, the combinations where the product is LESS OR EQUAL than the sum are:

1 and 1

1 and 2

1 and 3

1 and 4

1 and 5.

2 and 1

3 and 1

4 and 1

5 and 1

2 and 2.

10 combinations.

Then the combinations where the product is larger than the sum is:

25 - 10 = 15.

Then the probability that we are looking for is:

P = 15/25