It's even more non-intuitive if you think about both this one and the sleeping beauty "paradox" at the same time.

However, in both cases you have to be really careful with how you lay out the problem. Slight differences in wording can result in the opposite answer being right, as you've found out from Konrad doing what you showed in the picture instead of what you meant.

Because, of course, if there are three boxes, and you open the first drawer in your box, as you say, it matters whether you live in a universe with a gold-silver box or a silver-gold box. Your picture shows the gold-silver universe. In that universe, if you pick the first drawer, you have a 2/3 chance of getting gold the first time, and if you got gold, a 1/2 chance of getting gold the second time. In the silver-gold universe, getting gold from the first drawer gives you a 100% chance of getting gold the second time (because only the gold-gold box has a left-gold).

Now, if the placement of the coins was random--so the gold-silver and silver-gold universes are equally likely, but once we know you got a gold, you have a 1/3 chance of living in the silver-gold universe and a 2/3 chance of living in the gold-silver universe. Then everything works out as you said: (1/3 * 1) + (2/3 * 1/2) = 1/3 + 1/3 = 2/3.

The juxtaposition with the sleeping beauty paradox is especially tricky if you use the sleeping beauty version that works out to 1/2 and the box paradox that works out to 2/3.

It's a good lesson that one has to be really careful when dealing with correlated probabilities, and keep very careful track of what information is revealed by observations.