Size of a fair-coin test on 36 tosses

Question

To test whether a coin is fair you toss it $36$ times, count heads $X$ , and reject fairness whenever $|X-18|\ge4$ . What is the significance level $\alpha$ of this test, to three decimals?

Short answer

$0.243$ . Under a fair coin $X\sim\text{Bin}(36,\tfrac12)$ , and $\alpha=\mathbb{P}(X\le14)+\mathbb{P}(X\ge22)=2\,\mathbb{P}(X\le14)\approx0.243$ by symmetry.

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Populations, Samples, and Sampling Distributions
Sample Spaces, Events, and the Probability Axioms

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