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This is a variation of the famous Monty Hall problem.

I assume you know the usual setup. Here, the host behaves a bit different:

The host knows what lies behind the doors, and (before the player's choice) chooses at random which goat to reveal. He offers the option to switch only when the player's choice happens to differ from his.

Am I better off switching, or not?

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No, in this case, half the time you pick a goat, he doesn't give you the opportunity to switch. So when you are given that opportunity, the probability that switching is good ends up being 1/2. –  Thomas Andrews Feb 15 at 16:56

1 Answer 1

One third of the time the initial choice the player makes is the prize and Monty offers a change

One third of the time the initial choice is Monty's Goat - no goat is shown

One third of the time the initial choice is the other goat - a goat is shown ad a choice is given

In two cases you are shown a goat - one of these you have picked the prize, the other you have a goat. It's $50:50$ stick or change.

In the third case you are not shown a goat. By this you know you have picked a goat (!) and you should change (if that is on offer) for a fifty percent chance of a win.

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Last line is potentially confusing. The question states: "He offers the option to switch only when the player's choice happens to differ from his." So he clearly does not offer a chance to change in this case. –  Thomas Andrews Feb 15 at 18:27
    
@ThomasAndrews I wasn't sure what was intended - it is very different from the original where a goat is always shown and switch is always offered. Of course if know goat is shown you know you've lost if you can't switch - but there's no drama in that. –  Mark Bennet Feb 15 at 18:42

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