The Average p-value of a True Null

A fair coin is flipped 4 times and the number of heads $X$ is recorded. The one-sided exact p-value for testing $H_0:p=1/2$ against $H_1:p>1/2$ is $p(X)=P(\text{Binomial}(4,1/2)\ge X)$. When $H_0$ is true, what is the expected value $E[p(X)]$? Give a reduced fraction.

• distributions
• expected value
• hypothesis testing

Source: Original problem written for AxiomIQ. Related to the known fact that discrete p-values are stochastically larger than uniform.