I have seen this before, but didn't buy it then either.

Yes, we have a 50% chance of being right if we change our selection, but we also have a 50% chance of being right by sticking with the original answer. This is because this is a new decision that has no dependence on the first decision. We were not required to predict a sequence of events, but are making two distinct decisions.

So where is the advantage?

The typical explanation is that because we had a 33% chance of being right when we made our first decision, then this is still our probability if we don't change. This is mathematics mumbo-jumbo. The math isn't wrong, it is just that the math is using the wrong constraints and assumptions. We we are making a second decision with a new set of constraints, so what difference does it make what our

odds were before we make the second decision?