# p-value definition and inadequacy of tail events [duplicate]

Wikipedia defines p-values as

(The probability of obtaining a test statistic) equal to or more extreme than what was actually observed

For starters, usage of extreme there is subjective. To make matters more concrete,

$$Pr ( X ≥ x | H )$$ for right tail event
$$Pr ( X ≤ x | H )$$ for left tail event
$$2 min \{ Pr ( X ≤ x | H ) , Pr ( X ≥ x | H ) \}$$ for double tail event.

This diagram explains it for a right tailed event:

Now these definitions work well for normal distribution, t distribution, F distribution, and chi-square distributions.

But consider a bimodal distribution.

     ⏜       ⏜
/   \    /   \
/     \_/     |\
/              | \
x


For the observation x, won't we also have to consider the middle region as "extreme values" since their probability is low? So our extreme region will look something like the shaded region in:

     ⏜       ⏜
/   \    /   \
/|    |\_/|   |\
/.|    |...|   |.\
x


In this case we can't just consider tailed events. What exactly is the definition of "extreme values" then?