First, I will preface this question with my ulterior motive: I would like more evidence that the use of 19th and 20th century approximations play little to no pedagogic advantage in modern intro stats or intro data science courses.
First, let us agree to work with the following definition of a P-value: The probability of observing your sample—or something more extreme—given that the null hypothesis is true.
We wish to conduct a two-tailed hypothesis test for a population proportion using counts and exact probabilities from the binomial distribution. The hypotheses are $$H_0 : p = 10\%$$ $$H_a : p \ne 10\%$$ The sample obtained has $n=189$ and there are $k=10$ successful observations in this sample.
¿What is the two-tailed P-value for this test? It seems that there are reasonable arguments for either $$P(X \le 10) + P(X \ge 27) = 0.053$$ or $$P(X \le 10) + P(X \ge 28) = 0.038$$ (For those of you "addicted" to the conventional significance level of $\alpha=0.05$, you can probably see where I might be going with this. ;-)
To keep this in a pedagogic framework, I'm most curious for answers that might indicate how you would grade a student's work who submitted either answer...and how you would justify any loss of points that might occur.