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Probability that X<=Y when X and Y are p values from two different hypotheses

If i test one truly alternative hypothesis with 95% power, and one truly null hypothesis, what is the probability that the the p value from the truly alternative hypothesis test will be less than the p value from the truly null hypothesis test?

If i test two hypotheses, one of which has a null that is -in fact- false and the other has a null that is -in fact- true, I want to know the probability that the first will test will obtain a p value less than that second.

I can simulate the situation in R, and come up with a reasonable estimate that way, but I want to know how I can impute the answer exactly.

Using the program R:

a=vector()
b=vector()
for (i in 1:1000) {
+ ai = rnorm(108, mean = 4, sd = 20)
+ a = c(a,t.test(ai)p.value)
+ bi = rnorm(108)
+ b = c(b,t.test(bi)\$p.value)
+ }
q=rep(0,length=length(a))
q[a less than b]=1
mean(q)
[1] 0.844

So, I find that approximately 85% of the time, the truly alternative hypothesis yields a lower p value, but how can I impute this result. I want to prove it mathematically.