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I am on the Binomial Test entry in Wikipedia and in it, there is an example where a die is rolled 235 times, and 6 comes up 51 times. They claim there are two ways to construct a two-sided hypothesis test to compute a p-value for this.

There are two methods to define the two-tailed p-value. One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value. The probability of that occurring in our example is 0.0437. The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions.

I understand what the second method is, and that is the only method I am familiar with. That is, define the p-value to be

$$ \displaystyle P_{val} = \sum_{p(y) \leq p(51)}p(y) $$

This means summing up the probabilities smaller or equal to that of the event of 6 coming up 51 times.

However, I am lost what the first method is. What does it mean, "sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value."?

This seems like it would imply summing probabilities from 51 to 235, then summing from 1 to 50, which is obviously incorrect. What are they trying to say?

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    $\begingroup$ "... sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value" is awful English. Given an observation $x$ (treated now as a number, not a random variable) and an expected value of $\mu$ under the null hypothesis, it's trying to refer to the probability of the event $|Y-\mu|\ge|x-\mu|$ when $Y$ follows the null distribution. $\endgroup$
    – whuber
    Aug 6 at 13:19

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