I have three coins, like the coins from

Fifty-fifty. The first coin is a regular coin with heads and tails sides; the second coin has heads on both sides; the third coin has tails on both sides.

Two coins are chosen and flipped, with the third still shrouded in a pouch.
The results of the flips shows one head and one tail. What is the probability that the normal coin is the coin still in the pouch?

Yes, this time it is fifty-fifty.

I used bayes reasoning:

A priori, there are three possibilities:

a) Good coin is in the bag, and the faces necessarily differ:

Probability = 1/3

b) Good coin is not in the bag, and the faces differ

Probability = 2/3 * 1/2 = 1/3

c) Good coin is not in the bag, and the faces agree

Probability = 2/3 * 1/2 = 1/3

We can rule out case (c). Having done so, the good coin is still in the bag in half of the cases, so probability = 50%.