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Evidently the game favors Bob, but for the given $p,q$, what is the choice of $n$ that maximizes Alice's chances of winning? The problem of determining the optimal $N$ first appeared in \\cite{wa}. We show that there is an essentially unique value $N(q,p)$ of $n$ that maximizes the probability $f(n)$ that the weak coin will win, and it satisfies $\\frac{1}{2(p-q)}-\\frac12\\le N(q,p)\\le \\frac{\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1002.1763","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-02-09T03:15:46Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"84453e0960c15fd82303f4feaeba78331b6b09f309f5192d2d290a3dc9605f9e","abstract_canon_sha256":"36107a5edc21f23c457c830e4ff5134c8c122d9bdb767825c52a98098a992056"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:39.029663Z","signature_b64":"7CpsaYTMVzpxl85aNZICtJLoKKuNvpuKSVDaSzQaKSLuWQy4RKq+HwslVzwYut5pBw7Fsk1TXCNPPg1dSAKWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7361f2c6cbf7e0ac3bac17ab1cbc58ec6bb70ea5230f38f4ac3e62c1d3c6a1f1","last_reissued_at":"2026-05-18T02:24:39.029094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:39.029094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How to lose as little as possible","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Herb Wilf, Stan Wagon, Vittorio Addona","submitted_at":"2010-02-09T03:15:46Z","abstract_excerpt":"Suppose Alice has a coin with heads probability $q$ and Bob has one with heads probability $p>q$.\n  Now each of them will toss their coin $n$ times, and Alice will win iff she gets more heads than Bob does. 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