Switching is actually the best strategy, with you winning the car 2/3 times if you always switch.
"But wait!" you say "there's only two outcomes, so there's a 1/2 chance." No, not quite. Just because there are two outcomes doesn't mean they're are equally likely.
"Hang on!" you say "Removing a door with a goat behind it means that there's one door with the car, and one door without, so it should be 1/2 chance. It's like the revealed goat was never there." Not quite right again, the removed goat does matter, and the fact that you started with it affects the probability.
There are a few different ways to explain this, and different ones work best for different people, so click through the links below to find an explanation that works best for you.
When you are done looking at these explanations, you can click this link to see more explanations and videos. Or you can view the bibliography.
*All the explanations do involve math, but the first two use visuals or other thought experiments more.
(C) Zoe Hills, 2016