Testing Whether a Coin Is Fair

Problem A coin shows 550 heads out of 1000 flips. How would you determine whether it is fair?

Be ready to discuss

• Framing as a hypothesis test: null hypothesis p = 0.5 (fair), two-tailed alternative p ≠ 0.5 — and why two-tailed is the right default unless you had a prior reason to suspect bias toward heads specifically.
• The test statistic: z = (p̂ − 0.5) / sqrt(0.25/n). With p̂ = 0.55 and n = 1000 the standard error is ≈ 0.0158, giving z ≈ 3.16 and a two-tailed p-value ≈ 0.0016 — significant at any conventional threshold, so you would reject fairness.
• The confidence-interval view: 0.55 ± 1.96 × 0.0158 ≈ (0.519, 0.581), which excludes 0.5 — the same conclusion expressed as a range, and generally the more informative presentation.
• Why the normal approximation is valid here (np and n(1−p) both far above 10), and what you would use instead for small n — an exact binomial test.
• Signal versus noise and sample size: the same 55% on 100 flips would not be significant; discuss the power of the test and how many flips you would need to detect a given bias.
• Statistical versus practical significance: a real but tiny bias may be detectable yet irrelevant, and repeated testing until you get significance is p-hacking.
• Bonus framing: the Bayesian alternative — a Beta prior updated to a Beta(551, 451) posterior, and reporting the credible interval instead.