{0,1,3,4,5,9}

So we need two of these to add to a multiple of 11, since 9+9=18 is the largest sum the only possible multiples are 0 and 11.

The only way to get 0 is with 0+0 and there is no pair of values that add to 11.

Thus the only possible way for a^2+b^2 to be divisible by 11 is if both a and b are also divisible by 11.  Since each of these have a 1/11 chance of being divisible by 11 then the total probability is (1/11)^2 or 1/121