Sure. Let H0 be the null hypothesis and D be the data you observed. The first statement is P(H0|D) = 0.05. The second is P(D|H0) = 0.05.
The two quantities are related to each other via Bayes rule:
P(H0|D)=P(D|H0)P(H0)/P(D)
So indeed, as P(D|H0) goes down, so does P(H0|D). But if P(H0)/P(D) is sufficiently large, you can easily have P(H0|D) high while P(D|H0) is low.
I too have sympathy for everyone confused by frequentist stats - they tend to answer the exact opposite question that one really wants answered. In contrast, Bayesian stats tend to answer the question that most people ask.
P(D) is the probability of observing the data you just saw, due to either the null or non-null hypothesis. It's a strictly Bayesian quantity, since it's dependent on a prior. If your model has only a null and alternative hypothesis, then:
The two quantities are related to each other via Bayes rule:
P(H0|D)=P(D|H0)P(H0)/P(D)
So indeed, as P(D|H0) goes down, so does P(H0|D). But if P(H0)/P(D) is sufficiently large, you can easily have P(H0|D) high while P(D|H0) is low.
I too have sympathy for everyone confused by frequentist stats - they tend to answer the exact opposite question that one really wants answered. In contrast, Bayesian stats tend to answer the question that most people ask.