My idea is as follows(I try to avoid some jargons in Bayesian probability theory).
There are 3 balls, A,B,C. The guest picks up one ball, and then host announces one "bad" ball from the remaining two balls since he knows where is the true ball. I will suppose A is the right ball.
After the actions of the guest and the host, which state of the world the guest is in? Image three balls are in a pool, and the guest picks up one from the pool. We shall have:
1. the guest picks up A, and leaving B,C in the pool.
2. the guest picks up B, and leaving A,C in the pool.
3. the guest picks up C, and leaving A,B in the pool.
Let's call these states as (B,C), (A,C), and (A,B) respectively. We have the probability that the guest is in each state is 1/3.
Now, the host names one "bad" ball from the pool, and marks it so that the guest can recognize the bad from the remaing two balls in the pool.
Not Switch case:
Suppose the guest does not switch his choice. If the guest is in (B,C) states, then the conditional probability he picks up the right ball is 1. If the guest is in (A,C) or (A,B) state, then the conditional probability he picks up the right ball is 0. Hence the probability he picks up the right ball is (1/3)*1+(1/3)*0+(1/3)*0=1/3
Switch case:
Suppose the guest switches his choice. If the guest is in (B,C) states, then the conditional probability he picks up the right ball is 0 since any ball in the pool is bad. If the guest is in (A,C) or (A,B) state, then the conditional probability he picks up the right ball is 1 since the only bad ball in the pool is marked. Therefore, the probability he picks up the right ball is (1/3)*0+(1/3)*1+(1/3)*1=2/3, and the probability he picks up the bad ball is (1/3)*1+0+0=1/3.
Bayesian Theorem
Define
event Ci=the ball i is the right ball.
event Hij=the host marks ball j after the guest picks up ball i
event CiHik=conditional on the host marks ball j after the guest picks up ball i, the ball i is the right ball.
Our target
Not switch:Pr(CiHik)
Swithtch: Pr(CjHik)
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