[pdurrant]

Originally Posted by obs20

Here's my try.

Spoiler:

This is a very bad bet to begin with. The payoff is even money for an event that only has a 1/3 chance of occurring. Rule of thumb we only increase the wager if the odds become better than even money.
In the first instance one of the cups was over turned which were of the host's choosing this shouldn't change the odds at all. The host had insider knowledge. much. The odds are still 1/3 that we made the right choice originally but now faced with a new choice of staying or switching it is still a choice of 1/2.
However if one were to overturned by chance and the odds of that the cup on the left be overturned and that it is empty now the chance of 2 cups being overturned and both being empty is 1/6( 1/3 * 1/2) so now the odds are 5/6/2 that I've picked the right one cup. The odds are now 5:12 . Hold the bet.

Your reasoning is correct up until the second half of the fourth sentence of your second paragraph.

Spoiler:

Right:
The odds are still 1/3 that we made the right choice originally

Wrong:
but now faced with a new choice of staying or switching it is still a choice of 1/2