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Nineteen tails in a row may still be due to chance? | blogjou

Daniel S. Milo

Good Enough - The Tolerance for Mediocrity in Nature and Society

p. 133
04 Nov 2023

Nineteen tails in a row may still be due to chance?


The textbook example of the null hypothesis is the flip of a coin. Each coin flipped has a fifty-fifty chance of landing on heads or tails. According to the lat of large numbers, one has to repeat the flip many times in order to achieve the expected 1:1 ratio of heads to tails. How many times? The consensual number, twenty, was theorized by the statistician an geneticist Ronald Fisher. According to Fisher’s significance level, nineteen tails in a row may still be due to chance, but twent signifies a loaded coin. In experimental science, results obtained twenty times are said to be signfiicant, refuting the null hypothesis.

Something is wrong her with the math, isn’t it? Under the null hypothesis, the probability of having nineteen tails in a row is $(1/2)^{19}$, which is much smaller than $0.05$. So we would already reject the null hypothesis. Actually $2^5=32>20$, five tails in a row would be enough to reject the null hypothesis. That does not sound right either.