Pythagorean odds

Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform distribution. The square root of the sum (k·a+1)² + (k·b+1)² is then computed and rounded to the nearest integer. If the result is equal to k, he scores k points; otherwise he scores nothing.

For example, if k = 6, a = 0.2 and b = 0.85, then (k·a+1)² + (k·b+1)² = 42.05. The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6. This is equal to k, so he scores 6 points.

It can be shown that if he plays 10 turns with k = 1, k = 2, ..., k = 10, the expected value of his total score, rounded to five decimal places, is 10.20914.

If he plays 10⁵ turns with k = 1, k = 2, k = 3, ..., k = 10⁵, what is the expected value of his total score, rounded to five decimal places?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter