\(5\) izinombolo ezilinganayo ziholela eqhinga. Kodwa-ke, ukuthi lokhu kungenzeka kangakanani kuncike kulokho okubala njengokuzama: umqulu owodwa \(5\) wamadayisi noma ukujika okuphelele lapho ugcina khona ukufanisa amadayisi futhi uphinde uwagoqe kuze kufike \(2\) izikhathi. Lapha sisebenzisa \(5\) idayisi elilungile elinezinombolo \(1\) kuya ku \(6\) kanye nemiphumela ezimele. Wonke amaqhinga abalwa njengempumelelo; amathuba owafunayo awakhawulelwe enanini elithile lamaphoyinti afana nokuthi \(6\) .
Ngokwemiyalelo yomdlalo , unganquma futhi ngaphambi komqulu ngamunye owengeziwe ukuthi yimaphi amadayisi ashiywe elele. Umgomo wethu kuphela uhlobo oluthile lweqhinga, hhayi amaphuzu angcono kakhulu kugeyimu. I-roll eyodwa inemiphumela \(6^5=7776\) cishe efanayo lapho uhlukanisa idayisi. Kulezi \(6\) bonisa inombolo efanayo kuwo wonke amadayisi. Ngakho-ke ithuba lithi \(\frac{6}{7776}=\frac{1}{1296}\) . Ngokushesha nje lapho iqhinga selifinyelelwe, uqeda umnyakazo; Ukuphonswa okuphezulu \(3\) kuvunyelwe sekukonke.
$$p_1=\frac{6}{6^5}=\frac{1}{1296}\approx0{,}07716\%.$$
Isu elincane kakhulu futhi lingcono kakhulu: uma ufuna ukwandisa ithuba, ugcina iqembu elikhulu lezinombolo ezifanayo. Inombolo eqondiwe ingaphinda ishintshe: Kusukela ku \((2,2,3,4,5)\) uqala ngokugcina okubili \(2\) . Uma umqulu olandelayo uletha \((6,6,6)\) , ugcina leli qembu bese ugoqa kabusha kokubili \(2\) . Ithuba lokugcina liyakhuphuka lisuka ku \(\frac{1}{216}\) liye ku \(\frac{1}{36}\) . Okokugcina, ake sicabange: I-gimmick iqeda isiqephu ngokushesha; amanye ama-litter awafakiwe.
Indlela elula yokwenza ukubala ukuqala ngokuphonswa kokugcina. Uma \(4\) wedayisi elifanayo selivele litholakala, idayisi elisele kuphela okufanele lidlule: ithuba lithi \(\frac{1}{6}\) . Uma \(3\) kuyidayisi elifanayo, amanye amabili kufanele abonise inombolo ebanjiwe. Ngenxa yokuthi imiphumela yabo izimele, phindaphinda amathuba futhi uthole \(\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}\) . Ngakolunye uhlangothi, uma ubambelela ezinombolweni ezahlukene ngesikhathi esisodwa, angeke kube khona inkinga ngokuphonswa kokugcina. Ngakho-ke akuyona nje inombolo ebalulekile, kodwa nokuthi yini oyigcinayo.
Ngokusele \(2\) kwamaroli, ukufa ngakunye okulahlekile kuthola elinye ithuba. Uma \(4\) idayisi ifana, ukushaya okukodwa ngedayisi elisele kwanele. Isibalo sekhawunta sifushane lapha: Ugeja inombolo eqondiwe kuyo yomibili imizamo namathuba okuthi \(\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36}\) . Ngakho-ke ithuba lokuphumelela lithi \(1-\frac{25}{36}=\frac{11}{36}\) . Emaqenjini amancane, yonke imiphumela engaba khona yokuphosa ibalwa futhi okuqondiwe kungashintshwa. Amathuba alandelayo avela ngokukhetha okungcono kakhulu kokuma:
| Iqembu elikhulu kunawo wonke | Imiphumela kudoti \(1.\) | Ithuba \(2\) namaroli asele |
|---|---|---|
| \(1\) | \(720\) | \(\frac{221}{17496}\) |
| \(2\) | \(5400\) | \(\frac{113}{3888}\) |
| \(3\) | \(1500\) | \(\frac{121}{1296}\) |
| \(4\) | \(150\) | \(\frac{11}{36}\) |
| \(5\) | \(6\) | \(1\) |
Imigqa ayenzeki ngefrikhwensi efanayo. Isibonelo, iqembu elikhulu \(2\) lamadayisi liyakhiwa \(5400\) \(7776\) wokuphonswa kokuqala okungaba khona. Lokhu kufaka phakathi ipheya eyodwa kanye \(2\) namapheya; Uma kukhona uthayi, ukhetha eyodwa yazo. Siphindaphinda ithuba ngalinye kwesokudla ngenombolo ephakathi futhi sihlukanise isamba ngokuthi \(7776\) . Lokhu kubuyisela \(p_3\) ; kokuthi \(p_2\) usebenzisa kuphela ukuphonswa okusele. Amanani alandelayo asebenza ngaphambi kokuphonswa kokuqala okunenketho enkulu yokuma: \(p_2\) ngobuningi obungu \(2\) kanye no \(p_3\) obungangeni \(3\) wokuphonswa:
$$p_2=\frac{221}{17496}\approx1{,}26315\%.$$
$$p_3=\frac{347897}{7558272}\approx4{,}60286425\%.$$
Ukuthi ukugcina iqembu elikhulu kunawo wonke kungcono kakhulu kungahlolwa ngokugcwele. Umbhalo we-PHP olandelayo ucubungula konke okukhethwa kukho, okuhlanganisa izinombolo ezixubile noma awekho amadayisi aphethwe nhlobo. Iqala ekupheleni kwejika, bese ibala izinqumo ezingcono kakhulu \(1\) kanye \(2\) namaroli asele futhi ekugcineni ibe okumaphakathi ngaphezu komqulu wokuqala. Amaphethini afanayo aphinde asetshenziswe ukugcina umthethosivivinywa omncane. Konke ukuqhathanisa kwenziwa ngamanani aphelele; isibonisi samaphesenti kuphela esiyindilinga:
<?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);
Cishe \(4{,}60\%\) kusho: Ngisho nokukhetha okungcono kakhulu, inkinga ihlala ingavamile. Uma ubhalisa kabusha yonke into ngaso sonke isikhathi, ithuba elingaphezu komkhawulo \(3\) wemizamo limayelana \(0{,}23130\%\) kuphela . Uma uqhubeka nokufanisa amadayisi, kodwa unamathele enombolweni eqondiwe ekhethwe ngemva komqulu \(1.\) , uzozuza cishe \(4{,}52941\%\) . Ukushintshela eqenjini elikhulu ngakho-ke kuletha inzuzo encane eyengeziwe. Kungakhathaliseki ukuthi iqhinga lezibalo noma umuzwa omsulwa wamathumbu: I-Kniffel iyigeyimu yebhodi engaphelelwa isikhathi, ejabulisayo futhi, ngaphezu kwakho konke, ejabulisayo.