I don’t agree:
This is a binomial probability problem. Each roll of a fair six-sided die is an independent trial with success probability \(p = \frac{1}{6}\) (rolling a 6) and failure probability \(1 - p = \frac{5}{6}\) (rolling anything else). We want the probability of exactly \(k = 5\) successes in \(n = 8\) trials.
The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
First, compute the binomial coefficient:
\[
\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!} = 56
\]
Substitute into the formula:
\[
P(X = 5) = 56 \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right)^3 = 56 \cdot \frac{1^5 \cdot 5^3}{6^8} = 56 \cdot \frac{125}{1,679,616} = \frac{7,000}{1,679,616}
\]
Simplify the fraction by dividing numerator and denominator by their greatest common divisor (which is 8):
\[
\frac{7,000 \div 8}{1,679,616 \div 8} = \frac{875}{209,952}
\]
This fraction is in lowest terms. As a decimal approximation, this is about 0.00417 (or 0.417%).
bert / Bert
npub1jq…gr247
2025-07-23 06:44:04 CEST
in reply to nevent1q…8mta
Author Public Key
npub1jqs0u7zhh53e94gyhm4eu458wm6sw7z0kk66jjhhkhh346tcq2ysfgr247Published at
2025-07-23 06:44:04 CESTEvent JSON
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"content": "I don’t agree:\n\nThis is a binomial probability problem. Each roll of a fair six-sided die is an independent trial with success probability \\(p = \\frac{1}{6}\\) (rolling a 6) and failure probability \\(1 - p = \\frac{5}{6}\\) (rolling anything else). We want the probability of exactly \\(k = 5\\) successes in \\(n = 8\\) trials.\n\nThe binomial probability formula is:\n\\[\nP(X = k) = \\binom{n}{k} p^k (1 - p)^{n - k}\n\\]\n\nFirst, compute the binomial coefficient:\n\\[\n\\binom{8}{5} = \\frac{8!}{5!(8-5)!} = \\frac{8!}{5! \\cdot 3!} = 56\n\\]\n\nSubstitute into the formula:\n\\[\nP(X = 5) = 56 \\left( \\frac{1}{6} \\right)^5 \\left( \\frac{5}{6} \\right)^3 = 56 \\cdot \\frac{1^5 \\cdot 5^3}{6^8} = 56 \\cdot \\frac{125}{1,679,616} = \\frac{7,000}{1,679,616}\n\\]\n\nSimplify the fraction by dividing numerator and denominator by their greatest common divisor (which is 8):\n\\[\n\\frac{7,000 \\div 8}{1,679,616 \\div 8} = \\frac{875}{209,952}\n\\]\n\nThis fraction is in lowest terms. As a decimal approximation, this is about 0.00417 (or 0.417%).",
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