\(5\) helu like ka hopena i ka maalea. Eia nō naʻe, e pili ana paha kēia i ka mea āu e helu ai ma ke ʻano he hoʻāʻo: hoʻokahi ʻōwili o nā dice \(5\) a i ʻole kahi huli piha āu e hoʻomau ai i ka hoʻohālikelike ʻana i nā dice a ʻōwili hou iā lākou a hiki i ka manawa \(2\) . Ma ʻaneʻi, hoʻohana mākou i \(5\) dice kaulike me nā helu \(1\) a i \(6\) a me nā hopena kūʻokoʻa. ʻO kēlā me kēia hoʻopunipuni i helu ʻia he kūleʻa; ʻaʻole i kaupalena ʻia ka probalika āu e ʻimi nei i kahi helu kikoʻī o nā kiko e like me \(6\) .
E like me nā ʻōlelo aʻoaʻo pāʻani , hiki iā ʻoe ke hoʻoholo hou ma mua o kēlā me kēia ʻōwili hou i waiho ʻia nā dice e wahaheʻe. ʻO kā mākou pahuhopu wale nō kekahi ʻano hana hoʻopunipuni, ʻaʻole ka helu maikaʻi loa i ka pāʻani. He \(6^5=7776\) ka hopena i ka wā e ho'oka'awale ai i nā dice. ʻO kēia mau \(6\) i ka helu like ma nā dice a pau. No laila, ʻo \(\frac{6}{7776}=\frac{1}{1296}\) . I ka hiki ʻana i kahi hoʻopunipuni, hoʻopau ʻoe i ka neʻe; ʻAe ʻia ka nui o nā kīloi \(3\) .
$$p_1=\frac{6}{6^5}=\frac{1}{1296}\approx0{,}07716\%.$$
ʻO ka hoʻolālā koʻikoʻi ka mea maikaʻi loa: inā makemake ʻoe e hoʻonui i ka manawa, mālama ʻoe i ka hui nui o nā helu like. Hiki ke hoʻololi i ka helu helu: Mai \((2,2,3,4,5)\) mālama mua ʻoe i nā \(2\) . Inā hāʻawi ka ʻōwili aʻe i \((6,6,6)\) , mālama ʻoe i kēia hui a ʻōwili hou i nā ʻelua \(2\) . Piʻi ka manawa hope mai \(\frac{1}{216}\) i \(\frac{1}{36}\) . A ʻo ka mea hope akā ʻaʻole ka mea liʻiliʻi loa, e manaʻo kākou: Hoʻopau koke kahi gimmick i ka hanana; waiho ʻia nā ʻōpala hou aku.
ʻO ke ala maʻalahi loa e hana i ka helu ʻana ʻo ia ka hoʻomaka me ka pahu hope. Inā loaʻa \(4\) o nā dice hoʻokahi, ʻo nā dice i koe wale nō ke hala: ʻo ka manawa \(\frac{1}{6}\) . Inā ʻo \(3\) nā dice like, pono nā mea ʻelua e hōʻike i ka helu i paʻa. No ke kūʻokoʻa o kā lākou hopena, e hoʻonui i nā manawa a loaʻa ka \(\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}\) . Ma kekahi ʻaoʻao, inā ʻoe e paʻa i nā helu like ʻole i ka manawa like, ʻaʻohe pilikia me ka pahu hope. No laila, ʻaʻole ka helu wale nō ka mea nui, akā ʻo ka mea āu e mālama ai.
Me \(2\) nā ʻōwili i koe, loaʻa i kēlā me kēia make i nalo kahi manawa hou. Inā like nā dice \(4\) , ua lawa ka pā hoʻokahi me nā dice i koe. ʻOi aku ka pōkole o ka helu helu ma ʻaneʻi: Ua hala ʻo ia i ka helu i ʻike ʻia ma nā hoʻāʻo ʻelua me ka probability \(\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36}\) . No laila, 'o ka manawa o ka holomua \(1-\frac{25}{36}=\frac{11}{36}\) . No nā pūʻulu liʻiliʻi, helu ʻoe i nā hopena kūpono a pau o nā kiola a ʻae i ka hoʻololi ʻana i ka pahuhopu. ʻO nā manawa e hiki mai ana me ka koho maikaʻi loa o ka hoʻomaha:
| Hui nui loa | Nā hualoaʻa ma ka \(1.\) . | Loaʻa me \(2\) nā ʻōwili i koe |
|---|---|---|
| \(1\) | \(720\) | \(\frac{221}{17496}\) |
| \(2\) | \(5400\) | \(\frac{113}{3888}\) |
| \(3\) | \(1500\) | \(\frac{121}{1296}\) |
| \(4\) | \(150\) | \(\frac{11}{36}\) |
| \(5\) | \(6\) | \(1\) |
ʻAʻole hiki ke laina me ka pinepine like. No ka laʻana, hana ʻia kahi hui nui loa o nā dice \(2\) ma \(5400\) o nā kiola mua loa o \(7776\) . Loaʻa kēia mau mea hoʻokahi pālua a me \(2\) pālua; Inā loaʻa kahi kaula, koho ʻoe i kekahi o lākou. Hoʻonui mākou i kēlā me kēia manawa ma ka ʻākau me ka helu ma waena a puʻunaue i ka huina me \(7776\) . Hoʻihoʻi kēia \(p_3\) ; no \(p_2\) hoʻohana wale ʻoe i ke koena kiola. Hoʻohana ʻia nā koina ma mua o ka hoʻolei mua ʻana me ke koho hoʻomaha maikaʻi loa: \(p_2\) no ka lōʻihi o \(2\) a me \(p_3\) no ka lōʻihi o \(3\) kiola:
$$p_2=\frac{221}{17496}\approx1{,}26315\%.$$
$$p_3=\frac{347897}{7558272}\approx4{,}60286425\%.$$
Hiki ke ho'āʻo piha ʻia ka mālama ʻana i ka hui nui loa. E noʻonoʻo ana ka palapala PHP ma lalo nei i kēlā me kēia koho i hiki ke koho ʻia, me nā helu hui ʻia a ʻaʻohe dice i paʻa. Hoʻomaka ia ma ka hopena o ka huli ʻana, a laila e helu i nā hoʻoholo maikaʻi loa no \(1\) a me \(2\) nā ʻōwili i koe a ma hope o ka ʻawelika ma luna o ka ʻōwili mua. Hoʻohana hou ʻia nā ʻano like no ka mālama liʻiliʻi ʻana i ka bila. Hana ʻia nā hoʻohālikelike a pau me nā helu helu; ʻo ka pākēneka wale nō ka hōʻike i hoʻopuni ʻia:
<?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);
Ma kahi o \(4{,}60\%\) ʻo ia hoʻi: ʻOiai me ka koho maikaʻi loa, kakaikahi ka pilikia. Inā ʻoe e hoʻohuli hou i nā mea a pau i kēlā me kēia manawa, ʻo ka manawa ma kahi o ka nui o nā ho'āʻo \(3\) ma kahi o \(0{,}23130\%\) wale nō. Inā hoʻomau ʻoe i ka hoʻohālikelike ʻana i nā dice, akā pili i ka helu i koho ʻia ma hope o ka ʻōwili \(1.\) , e loaʻa iā ʻoe ma kahi o \(4{,}52941\%\) . ʻO ka hoʻololi ʻana i ka hui nui e lawe mai i kahi pōmaikaʻi liʻiliʻi. ʻOiai ʻo ia ka hoʻolālā makemakika a i ʻole ka manaʻo o ka puʻuwai maʻemaʻe: ʻO Kniffel kahi manawa pau ʻole, leʻaleʻa a ma luna o nā mea āpau, pāʻani papa leʻaleʻa.