re: What are some inaccurate beliefs you heard from others?

Here's how I understand it:

If you aren't going to switch, you've got a 33% chance of picking the car on your first guess.

If you are going to switch, picking a goat (67% chance) on your first guess guarantees that you win the car, since Monty removes the other goat every time. Boom new car.

I have no idea if this is a valid explanation, but that's how I wrap my head around it.

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In the Monty Hall problem you should always switch.

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I know this is true, but I can't understand why to save my life.

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It helps if you pretend there are 100 doors to start. You choose one, he then eliminates all the wrong ones and leaves just 2 (the one you picked and one more). There is a pretty good shot that you did not pick the correct door from the beginning, so it's obvious that the other door is the correct one.

Same concept when you only start with 3 doors, but the odds are just a little better that you would choose the correct one from the beginning (33% chance vs 1%).