Distribution of p-values
In any hypothesis test,
The diagram below shows typical distributions that might be obtained.
P-values and probability
P-values have a rectangular distribution between 0 and 1 when H0 holds. A consequence of this is that the probability of obtaining a p-value of 0.1 or lower is exactly 0.1 (when H0 holds). This is illustrated on the left of the diagram below.
Similarly, the probability of obtaining a p-value of 0.01 or lower is exactly 0.01, etc. (when H0 holds).
P-values are more likely to be near 0 than near 1 if the alternative hypothesis holds