Question 323342
There are 2 balls chosen. Thus, there are 10^2 = 100 possible outcomes.

Half of them sum to odds and half to evens.

Perhaps you have not dealt with generating functions, but we can illustrate it with them.

{{{(sum(x^k,k=1,10))^2}}}

If we expand this out, we get

{{{x^20+2x^19+3x^18+4x^17+5x^16+6x^15+7x^14+8x^13+9x^12+10x^11+9x^10+8x^9+7x^8+6x^7+5x^6+4x^5+3x^4+2x^3+x^2}}}

The coefficients represent the number of ways that each sum can occur. i.e. how many ways can we get a sum of 15?. Look at the coefficient of the x^15 term.
It is 6. There are 6 ways to get a sum of 15. How many ways can we get a sum of 20?. Look at x^20 and we see its coefficient is 1. There is 1 way to get a sum of 20, and that is by drawing two 10's.
Since there are 100 possible outcomes, count up the coefficients of the even exponents. They sum to 50. Therefore, the probability is {{{50/100=1/2}}} of the two balls summing to an even number. The other half sums to odd numbers.