47fc3c6761
* fix: clean-up Project Euler 281-300 * fix: missing image extension * fix: missing power Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing subscript Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
47 lines
1.3 KiB
Markdown
47 lines
1.3 KiB
Markdown
---
|
|
id: 5900f48a1000cf542c50ff9c
|
|
title: 'Problem 285: Pythagorean odds'
|
|
challengeType: 5
|
|
forumTopicId: 301936
|
|
dashedName: 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`.
|
|
|
|
```js
|
|
assert.strictEqual(pythagoreanOdds(), 157055.80999);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function pythagoreanOdds() {
|
|
|
|
return true;
|
|
}
|
|
|
|
pythagoreanOdds();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|