Problem 285
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?
毕达哥拉斯奇数
阿尔伯特选择了一个正整数k，然后在区间[0,1]内按照均匀分布随机选择了两个实数a和b。
然后，计算出(k·a+1)2 + (k·b+1)2的平方根，并四舍五入到最近的整数。如果结果等于k，他将获得k分；否则不得分。
例如，如果k = 6，a = 0.2，b = 0.85，那么(k·a+1)2 + (k·b+1)2 = 42.05。
42.05的平方根为6.484…，当四舍五入到最近整数时结果是6。
结果等于k，因此他得6分。
可以看出，如果他进行10轮，分别选择k = 1、k = 2、……、k = 10，他的期望总得分四舍五入到五位小数是10.20914。
如果他进行105轮，分别选择k = 1、k = 2、k = 3、……、k = 105，他的期望总得分四舍五入到五位小数是多少？