In the article Bayes or not Bayes, is this the question?, in the paragraph before Figure 2, the author says:
Let us suppose that we want to investigate whether the sex ratio in hypothetical mice population is 1:1. We can create two experimental designs. In the first experiment, we can randomly select a mouse until the first male is chosen. The result in this experiment is the total number of mice chosen. In the second experiment, we can randomly select exactly seven mice. The result of this experiment will be the number of male and female mice in a sample of seven. Let us suppose that the result was FFFFFFM. If we do not know what experimental design was used, this result is useless. In the first experiment, the P is 0.031, but in the second experiment, the P value is 0.227.
Checking this, assuming a null hypothesis of equal probability of male or female mice, the first should be the probability of needing $m\ge7$ trials to obtain a success (inclusive of success trial), and from the CDF:
$$ P_1 = 0.5^{m-1}\times 0.5 = 0.0039 $$
Thus the probability of needing $\ge 7$ trials under the null hypothesis is 0.0039 (0.39%).
While the second should be a binomial probability of $k$ successes in $N$ trials, with $k=0,1;N=7$, given by
$$ P_2 = \sum_{k=0}^{1} \binom{N}{k} 0.5^k 0.5^{N-k} = \binom{7}{1} 0.5^7 + \binom{7}{0} 0.5^7 = 0.063 $$
Where the probabilities $p=q=0.5$ under the null hypothesis. Thus the probability of pulling $\le 1$ male mice in a sample of 7 under the null hypothesis is 0.063 (6.3%).
But these are very far from the P-values in the article (0.0039 instead of 0.031, 0.063 instead of 0.227). Are these probabilities not the same as the author's intended p-values?
