\(5\) amanani alinganayo abangela iqhinga. Nangona kunjalo, ukuba oku kunokwenzeka kangakanani kuxhomekeke kwinto oyibala njengezama: umqulu omnye we- \(5\) idayisi okanye ujiko olupheleleyo apho ugcine ungqamanisa idayisi kwaye uwasonge kwakhona ukuya kuthi ga \(2\) amaxesha. Apha sisebenzisa \(5\) idayisi ezifanelekileyo ezinamanani \(1\) ukuya \(6\) kunye neziphumo ezizimeleyo. Yonke iqhinga libalwa njengempumelelo; ukuba kunokwenzeka okhangelayo akuphelelanga kwinani elithile lamanqaku afana \(6\) .
Ngokwemiyalelo yomdlalo , ungagqiba kwakhona phambi kokuba umqulu ngamnye owongezelelweyo ukuba yeyiphi idayisi esele ilele. Injongo yethu kuphela luhlobo oluthile lweqhinga, hayi awona manqaku angcono kumdlalo. Umqulu omnye uneziphumo \(6^5=7776\) ezinokuba neziphumo ngokulinganayo xa usahlula idayisi. Kwezi \(6\) bonisa inani elifanayo kuwo onke amadayisi. Ke ithuba lithi \(\frac{6}{7776}=\frac{1}{1296}\) . Ngokukhawuleza ukuba iqhinga lifikeleleke, ugqiba ukuhamba; Lilonke, ubuninzi bokujula \(3\) buvumelekile.
$$p_1=\frac{6}{6^5}=\frac{1}{1296}\approx0{,}07716\%.$$
Esona sicwangciso sincinci sesona silungileyo: ukuba ufuna ukwandisa ithuba, ugcina elona qela likhulu lamanani afanayo. Inani ekujoliswe kulo lingatshintsha kwakhona: Ukusuka \((2,2,3,4,5)\) kuqala ugcina ezimbini \(2\) . Ukuba umqulu olandelayo uhambisa \((6,6,6)\) , ugcina eli qela kwaye uphinde ukhuphe ezimbini \(2\) . Ithuba lokugqibela liyanyuka ukusuka \(\frac{1}{216}\) ukuya \(\frac{1}{36}\) . Kwaye okokugqibela kodwa okungakuncinananga, masicinge ukuba: I-gimmick iphelisa isiqendu ngokukhawuleza; ezinye iilitha azifakwanga.
Eyona ndlela ilula yokubala kukuqala ngokujula kokugqibela. Ukuba \(4\) yedayisi efanayo sele ikhona, kuphela idayisi eseleyo kufuneka idlule: ithuba ngu \(\frac{1}{6}\) . Ukuba \(3\) ngamadayisi afanayo, amanye amabini kufuneka abonise inani elibanjiweyo. Kuba iziphumo zabo zizimele, phinda-phinda amathuba kwaye ufumane \(\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}\) . Kwelinye icala, ukuba ubambelela kumanani ahlukeneyo ngexesha elinye, akunakubakho ngxaki ngokuphoswa kokugqibela. Ke ayisilo nani kuphela elibalulekileyo, kodwa nento oyigcinayo.
Nge \(2\) imiqulu eseleyo, ifa ngalinye elilahlekileyo lifumana elinye ithuba. Ukuba \(4\) idayisi ziyafana, ukubetha enye ngedayisi eseleyo kwanele. Ikhawuntala yokubala imfutshane apha: Uphosa inani ekujoliswe kulo kuzo zombini iinzame ngokunokwenzeka \(\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36}\) . Ithuba lokuphumelela ke ngoko \(1-\frac{25}{36}=\frac{11}{36}\) . Kumaqela amancinci, ubala zonke iziphumo ezinokubakho zokuphosa kwaye uvumele ukujoliswe kuko kutshintshwe. Amathuba alandelayo avela kunye nolona khetho lungcono lokumisa:
| Elona qela likhulu | Iziphumo kwi- \(1.\) litter | Ithuba nge \(2\) imiqulu eseleyo |
|---|---|---|
| \(1\) | \(720\) | \(\frac{221}{17496}\) |
| \(2\) | \(5400\) | \(\frac{113}{3888}\) |
| \(3\) | \(1500\) | \(\frac{121}{1296}\) |
| \(4\) | \(150\) | \(\frac{11}{36}\) |
| \(5\) | \(6\) | \(1\) |
Imigca ayenzeki ngamaxesha afanayo. Umzekelo, iqela elikhulu le \(2\) idayisi lenziwe kwi \(5400\) ye \(7776\) enokwenzeka yokuphosa okokuqala. Ezi ziquka iperi enye kunye \(2\) ngababini; Ukuba kukho iqhina, ukhetha enye yazo. Siphinda-phinda ithuba ngalinye ekunene ngenani eliphakathi kwaye sahlule isixa nge \(7776\) . Oku kubuya \(p_3\) ; ye \(p_2\) usebenzisa kuphela intsalela yokuphosa. La maxabiso alandelayo asebenza phambi kokujula kokuqala ngokhetho lokumisa olulolona: \(p_2\) ubuninzi be \(2\) kunye \(p_3\) kubuninzi be \(3\) ukujula.:
$$p_2=\frac{221}{17496}\approx1{,}26315\%.$$
$$p_3=\frac{347897}{7558272}\approx4{,}60286425\%.$$
Ukuba ngaba ukugcina elona qela likhulu kokona kulungileyo kungavavanywa ngokupheleleyo. Oku kubhaliweyo kwe-PHP kulandelayo kuthathela ingqalelo ukhetho olunokwenzeka, kuquka amanani axubeneyo okanye akukho dice libanjwe kwaphela. Iqala ekupheleni kojiko, emva koko ibala ezona zigqibo zingcono ze \(1\) kunye \(2\) nemiqulu eseleyo kwaye ekugqibeleni i-avareji ngaphezulu komqulu wokuqala. Iipateni ezifanayo ziphinda zisetyenziswe ukugcina ityala lincinci. Lonke uthelekiso lwenziwe ngee-integers; kuphela umboniso wepesenti ojikelezileyo:
<?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);
Ngokujikeleza \(4{,}60\%\) kuthetha: Nokuba ngolona khetho lungcono, ingxaki ihlala inqabile. Ukuba uphinda ubhalise yonke into ngalo lonke ixesha, ithuba elingaphezulu kobuninzi \(3\) imizamo imalunga \(0{,}23130\%\) kuphela. Ukuba ugcina idayisi ehambelanayo, kodwa unamathele kwinombolo ekujoliswe kuyo ekhethiweyo emva komqulu \(1.\) , uya kufezekisa ngeenxa zonke \(4{,}52941\%\) . Ukutshintshela kwiqela elikhulu ke ngoko kuzisa inzuzo encinci eyongezelelweyo. Nokuba sisicwangciso semathematika okanye imvakalelo emsulwa yamathumbu: I-Kniffel ayiphelelwa lixesha, iyonwabisa kwaye, ngaphezu kwayo yonke, umdlalo webhodi owonwabisayo.