What is the probability that the sum of the numbers from two rolled dice result in a perfect square or even number?

Part1: Finding probability of getting sum as a perfect square.

Maximum sum of both the dice is (6+6) equal to 12.

Up to 12, the perfect squares are: 1, 4 and 9.

Getting a sum of 1 from two dice is not possible. So, we are left with 4 and 9.

To get 4, the combination can be: (2,2) or (1,3) or (3,1). This means, to get the sum as 4, the probability is [3/36].

To get 9, the combination can be: (3,6) or (6,3) or (5,4) or (4,5). This means, to get the sum as 9, the probability is [4/36].

Therefore,the total probability of getting the sum as a perfect square is: [(3/36)+(4/36)]=[7/36].

Part2: Finding the probability of getting sum as an even number.

The possible even numbers can be 2, 4, 6, 8, 10 and 12. But, as 4 is already considered in part1, it should be ignored in this case.

The probability of getting sum as 2 is: [1/36]

The probability of getting sum as 6 is: [5/36]

The probability of getting sum as 8 is: [5/36]

The probability of getting sum as 10 is: [3/36]

By adding all the above, the probability of getting sum as an even number (ignoring 4) is:

[(1/36)+(5/36)+(5/36)+(3/36)]=[14/36].

From part 1 and part 2, we get the total probability as [(7/36)+(14/36)]=[7/12]=0.583333.