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Originally Posted by Jacklad
Think of it this way - I am Mary. I make my choice. What do I know? I know that I have a 1/3 chance of having the car. But I also know that there is a 2/3 chance that I don't. I also know with 100% certainty that at least one of the non-chosen ones is a goat. When that goat is revealed, it doesn't change anything that I already know - there is still a 2/3 chance that I didn't choose the car, so it's in my best interest to switch. Make more sense that way?
What makes this confusing is that the same logic applies to Susie, and it's in her best interest to switch as well. That's not impossible, because a 2/3 chance of success still implies a 1/3 chance of failure.
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Well at least you've given me my laugh for the day!
And I'll return the favor by conceeding that
if there is only one contestant, there is be a theoretical advantage in switching doors.
But in this case (where Monty opens the door that neither girl picked) each girl is left with a 50-50 chance of already having the right door. So there's a 50-50 chance she'll be wrong if she switches. In essense the switch question has become, "One of you has a car and the other has a goat, but I won't tell you which one. Would you like to switch?" That still looks like 50-50 to me.
Back to the original problem:
As presented on the tv show, there were two contestants who chose two different doors. As I see it, having two contestants makes a difference, and I think that's part of the reason we disagree about this problem. With two contestants, there's twice as much chance that one of them will pick the right door. (Dare I say there's a 2/3 chance one of them has chosen the right door?) So it could be equally well argued that after one contestant is eliminated, the remaining contestant has a better chance of winning by
not switching.
It's things like that that make me feel like my first impression was correct, but I think you've made a convincing enough case that there is at least a slight advantage in switching. Thank you for an interesting discussion. It's been fun.