{"type":"rich","version":"1.0","author_name":"bert (npub1jq…gr247)","author_url":"https://nostr.ae/npub1jqs0u7zhh53e94gyhm4eu458wm6sw7z0kk66jjhhkhh346tcq2ysfgr247","provider_name":"njump","provider_url":"https://nostr.ae","html":"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%)."}