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if n is even, n*n+1*n+2 is always going to be divisible by 8
example:
n = 2,n+1=3,n+2 = 4
since we have two consecutive even numbers in this, the product will be always divisible by 8..

try out numbers

4,5,6
6,7,8
8,9,10
14,15,16
18,19,20

2 will always be repeated three times..

so then the probability to get n as even would be :
48/96(half of them are even)
1/2

Does that help? maybe someone else can explain better

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