1.3 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f48a1000cf542c50ff9c | Problem 285: Pythagorean odds | 5 | 301936 | problem-285-pythagorean-odds |
--description--
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 {(ka + 1)}^2 + {(kb + 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 {(ka + 1)}^2 + {(kb + 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, \ldots, k = 10
, the expected value of his total score, rounded to five decimal places, is 10.20914.
If he plays {10}^5
turns with k = 1, k = 2, k = 3, \ldots, k = {10}^5
, what is the expected value of his total score, rounded to five decimal places?
--hints--
pythagoreanOdds()
should return 157055.80999
.
assert.strictEqual(pythagoreanOdds(), 157055.80999);
--seed--
--seed-contents--
function pythagoreanOdds() {
return true;
}
pythagoreanOdds();
--solutions--
// solution required