Question

1) Define p-values in the form of a conditional probability, i.e. p-value = Prob(A|B) where you say what A and B are
2) Define likelihood (from the likelihood ratio test) in the form of a conditional probability, i.e. likelihood = Prob(A|B) where you say what A and B are

Answer 1

1) The p value is defined as the probability of the observed values given that the null hypothesis is true. So we can define p value as a conditional probability. Define two events

A = random variable represents the observed data

B=the statistical hypothesis under consideration I.e the null hypothesis

Then P(A|B) =p value .

2) The likelihood can be expressed as

P(A|B) =likelihood function

A= Theta which is the parameter of the distribution

B= random variable

P is the pdf or pmf of the random variable.

Hence it is expressed as the conditional distribution.