\(5\) Tirooyinka isla eg waxay keenaan khiyaamo. Si kastaba ha ahaatee, sida ay tani u badan tahay waxay kuxirantahay waxaad tiriso iskuday ahaan: hal duub oo \(5\) laadhuu ah ama leexasho dhamaystiran oo aad ku sii wado laadhuuda isbarbardhiga oo dib u rogroga ilaa \(2\) jeer. Halkan waxaan ku isticmaalnaa \(5\) laadhuu cadaalad ah oo wata tirooyinka \(1\) ilaa \(6\) iyo natiijooyin madax-bannaan. Khiyaamo kastaa waxay tixgalinaysaa guul; itimaalka aad raadinayso kuma koobna tiro gaar ah oo dhibco ah sida \(6\) .
Sida ku cad tilmaamaha ciyaarta , waxaad go'aansan kartaa mar kale ka hor intaan duubin kasta oo dheeraad ah oo laadhuu ah oo jiifa laga tagay. Hadafka kaliya eeyagu waa nooc khiyaano ah, ma aha natiijada guud ee ugu fiican ciyaarta. Hal duub ayaa leh \(6^5=7776\) natiijooyin isku mid ah ayaa laga yaabaa marka la kala saaro laadhuuda. Kuwaas oo ah \(6\) waxay ku tusaysaa tiro isku mid ah dhammaan laadhka. Markaa fursaddu waa \(\frac{6}{7776}=\frac{1}{1296}\) . Isla markii khiyaamo la gaadho, waxaad dhammaynaysaa dhaqaaqa; Wadar ahaan, ugu badnaan \(3\) tuurid waa la ogolyahay.
$$p_1=\frac{6}{6^5}=\frac{1}{1296}\approx0{,}07716\%.$$
Xeeladda ugu yar ayaa sidoo kale ah tan ugu fiican: haddii aad rabto inaad sare u qaaddo fursadda, waxaad haysataa kooxda ugu badan ee tiro isku mid ah. Lambarka bartilmaameedku sidoo kale wuu bedeli karaa: Laga bilaabo \((2,2,3,4,5)\) marka hore waxaad ilaalisaa labada \(2\) . Haddii liiska soo socda uu keeno \((6,6,6)\) , waxaad haysaa kooxdan oo dib u rogaal celi labada \(2\) . Fursada u dambaysa waxay ka korodhaa \(\frac{1}{216}\) ilaa \(\frac{1}{36}\) . Ugu dambayn, laakiin aan ugu yaraan, aynu ka soo qaadno: Gimmick ayaa isla markiiba soo afjaraysa dhacdada; qashin kale ayaa laga tagay.
Sida ugu fudud ee xisaabinta loo samayn karaa waa in lagu bilaabo tuurista u dambaysa. Haddii \(4\) laadhuu isku mid ah la heli karo, kaliya laadhuu haray waa inay gudbaan: fursaddu waa \(\frac{1}{6}\) . Haddii \(3\) ay yihiin laadhuu isku mid ah, labada kale waa inay muujiyaan tirada la hayo. Sababtoo ah natiijadoodu waa madax bannaan, dhufo fursadaha oo hel \(\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}\) . Dhanka kale, haddii aad isku mar xajisato tirooyin kala duwan, wax dhib ah kuma dhici karto tuurista ugu dambeysa. Markaa ma aha tirada kaliya ee muhiimka ah, laakiin sidoo kale waxa aad hayso.
Iyadoo \(2\) duubabka haray, dhimasha kasta oo maqan waxay helaysaa fursad kale. Haddii \(4\) laadhuudu isku mid yihiin, hal ku dhufo laadhuu soo hadhay ayaa ku filan. Xisaabinta xisaabtu way ka gaaban tahay halkan: Wuxuu ku lumiyaa lambarka bartilmaameedka labada isku day ee suurtogalka ah \(\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36}\) . Haddaba fursada guushu waa \(1-\frac{25}{36}=\frac{11}{36}\) . Kooxaha yaryar, waxaad tirisaa dhammaan cawaaqibka suurtagalka ah ee tuurista oo u oggolow bartilmaameedka in la beddelo. Fursadaha soo socda ayaa la soo baxa doorashada ugu fiican ee joojinta:
| Kooxda ugu weyn | Natiijooyinka qashinka \(1.\) | Fursada \(2\) duubabka hadhay |
|---|---|---|
| \(1\) | \(720\) | \(\frac{221}{17496}\) |
| \(2\) | \(5400\) | \(\frac{113}{3888}\) |
| \(3\) | \(1500\) | \(\frac{121}{1296}\) |
| \(4\) | \(150\) | \(\frac{11}{36}\) |
| \(5\) | \(6\) | \(1\) |
Xariiqyadu kuma dhacaan inta jeer ee isku mid ah. Tusaale ahaan, kooxda ugu weyn ee \(2\) laadhuu ayaa lagu abuuray \(5400\) ee \(7776\) ee suurtogalka ah tuurista ugu horeysa. Kuwaas waxaa ka mid ah hal lamaane iyo \(2\) lammaane; Haddii uu jiro barbaro, waxaad doorataa mid iyaga ka mid ah. Waxaan ku dhufaneynaa fursad kasta oo dhanka midig ku taal lambarka dhexda ku yaal waxaana u qaybineynaa wadarta \(7776\) . Tani waxay soo celinaysaa \(p_3\) ; for \(p_2\) waxaad isticmaashaa oo kaliya tuurinta soo hadhay. Qiimayaashan soo socdaa waxay khuseeyaan ka hor tuurista kowaad oo leh doorasho joogsi fiican: \(p_2\) ugu badnaan \(2\) iyo \(p_3\) ugu badnaan \(3\) tuurista:
$$p_2=\frac{221}{17496}\approx1{,}26315\%.$$
$$p_3=\frac{347897}{7558272}\approx4{,}60286425\%.$$
In la haynta kooxda ugu weyn ay tahay mid si buuxda loo tijaabin karo. Qoraalka PHP ee soo socdaa wuxuu tixgelinayaa doorasho kasta oo suurtagal ah, oo ay ku jiraan tirooyin isku dhafan ama laadhuu aan la hayn gabi ahaanba. Waxay bilaabataa dhamaadka leexashada, ka dib waxay xisaabisaa go'aamada ugu fiican \(1\) iyo \(2\) duubabka hadhay iyo ugu dambeyntii celceliska ka sarreeya liiska hore. Nidaamyo isku mid ah ayaa dib loo isticmaalaa si biilka loo yareeyo. Dhammaan isbarbardhigga waxaa lagu sameeyay tirooyin; kaliya bandhiga boqolleyda ayaa wareegsan:
<?php
declare(strict_types=1);
if (PHP_INT_SIZE < 8) {
throw new RuntimeException('64-Bit-PHP erforderlich.');
}
function combinations(array $limits): array
{
$result = [[]];
foreach ($limits as $limit) {
$next = [];
foreach ($result as $prefix) {
for ($i = 0; $i <= $limit; $i++) {
$next[] = [...$prefix, $i];
}
}
$result = $next;
}
return $result;
}
function outcomes(int $number): array
{
static $cache = [];
if (isset($cache[$number])) {
return $cache[$number];
}
$result = [];
foreach (combinations(array_fill(0, $number, value: 5)) as $dice) {
$counts = array_fill(0, count: 6, value: 0);
foreach ($dice as $face) {
$counts[$face]++;
}
$key = implode(',', $counts);
$result[$key] = ($result[$key] ?? 0) + 1;
}
return $cache[$number] = $result;
}
function pattern(array $counts): string
{
rsort($counts);
return implode(',', $counts);
}
$states = [];
$values = [];
foreach (outcomes(5) as $key => $count) {
$counts = array_map('intval', explode(',', $key));
rsort($counts);
$states[pattern($counts)] = $counts;
$values[pattern($counts)] = (int) ($counts[0] === 5);
}
// Gemeinsamer Nenner: 6^(5 * verbleibende Würfe).
for ($remaining = 1; $remaining <= 2; $remaining++) {
$next = [];
foreach ($states as $key => $counts) {
$best = 0;
foreach (combinations($counts) as $held) {
$number = 5 - array_sum($held);
$total = 0;
foreach (outcomes($number) as $rolled => $count) {
$after = array_map('intval', explode(',', $rolled));
foreach ($held as $face => $amount) {
$after[$face] += $amount;
}
$total += $count * $values[pattern($after)];
}
$best = max($best, $total * (6 ** (5 - $number)));
}
$next[$key] = $best;
}
$values = $next;
}
$numerator = 0;
$denominator = 6 ** 15;
foreach (outcomes(5) as $key => $count) {
$numerator += $count * $values[pattern(array_map('intval', explode(',', $key)))];
}
[$a, $b] = [$numerator, $denominator];
while ($b !== 0) {
[$a, $b] = [$b, $a % $b];
}
echo intdiv($numerator, $a), '/', intdiv($denominator, $a), PHP_EOL;
printf("%.10f%%\n", (100 * $numerator) / $denominator);
Agagaarka \(4{,}60\%\) macnaheedu waa: Xataa doorashada ugu fiican, dhibaatadu waa naadir. Haddii aad wax walba dib ugu celiso mar kasta, fursada in ka badan ugu badnaan \(3\) isku day waxay ku saabsan tahay kaliya \(0{,}23130\%\) . Haddii aad sii waddo isbarbardhigga laadhuuda, laakiin aad ku dhegto lambarka bartilmaameedka ee la doortay ka dib duubkii \(1.\) , waxaad gaari doontaa agagaarka \(4{,}52941\%\) . U beddelashada kooxda weyn waxay keenaysaa faa'iido yar oo dheeraad ah. Iyadoo aan loo eegin haddii ay tahay xeelad xisaabeed ama dareenka mindhicirka saafiga ah: Kniffel waa wakhti aan wakhti lahayn, madadaalo leh, oo ka sarreeya, ciyaarta guddiga madadaalada.