Something really amazing:
Imagine that you are a contestant on the classic television game show Let’s Make a Deal. Behind one of three doors is a brand-new automobile. Behind the other two are goats. You choose door number one. Host Monty Hall, who knows what is behind all three doors, shows you that a goat is behind number two, then inquires: Would you like to keep the door you chose or switch? Our folk numeracy—our natural tendency to think anecdotally and to focus on small-number runs—tells us that it is 50–50, so it doesn’t matter, right?
How Randomness Rules Our World and Why We Cannot See It: Scientific American
Amazingly, the answer is: Switch, because you will win 2/3
But why is this so?
If you choose the correct door first, the other door will be a wrong door. If you choose a wrong door first, the other door has to be correct, because the host can't show you the correct door, just one with a goat behind. Because of that, the probability that you will win is the same as the one that you will choose a wrong door first. And that is 2/3.
If you do this with 10 doors and the host shows you that eight of the nine left are wrong doors, the probability that you will win if you switch is 9/10. Really amazing.
Here you can try it out by yourself:
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