Two envelopes with a real prior

Two envelopes hold amounts in a ratio of $2$ to $1$, but now the amounts are drawn from a known distribution: the smaller amount $A$ is equally likely to be \10$or$$20$(each with probability$\tfrac12$), giving envelope pairs${10, 20}$or${20, 40}$. You pick an envelope at random and open it to find$X$ dollars.

For each amount $X$ you might see, should you switch? What is the overall expected gain from a policy of always switching?