Here's how I would explain the Monty Hall problem:
First, get the person to realize that switching doors will always flip your initial outcome. If you picked a losing door, switching gives you the winning one. If you picked the winning door, switching gives you a losing one. There is no winning-to-winning or losing-to-losing. If you switch, you will always get the opposite of whatever you originally picked.
Once this idea has been understood, ask them to consider the probability that your original choice was a losing door. Once they agree that it's 2/3 (2 losing doors out of 3 possible choices), you can reply "that's right, and if there's a 2/3 chance that your current choice is a losing door, then there's a 2/3 chance that you're in a situation where switching will give you the winning door. Therefore, the chance that switching will give you the winning door is 2/3!"
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Here's how I would explain the Monty Hall problem:
First, get the person to realize that switching doors will always flip your initial outcome. If you picked a losing door, switching gives you the winning one. If you picked the winning door, switching gives you a losing one. There is no winning-to-winning or losing-to-losing. If you switch, you will always get the opposite of whatever you originally picked.
Once this idea has been understood, ask them to consider the probability that your original choice was a losing door. Once they agree that it's 2/3 (2 losing doors out of 3 possible choices), you can reply "that's right, and if there's a 2/3 chance that your current choice is a losing door, then there's a 2/3 chance that you're in a situation where switching will give you the winning door. Therefore, the chance that switching will give you the winning door is 2/3!"