The Monty Hall problem is quite tricky. I hope that you managed to follow the explanation and if you did well done. You may have also discovered that there are many ways of finding the correct mathematical result. When I looked at probability in the 'so you share a birthday' blog, I wrote 'the probability of an event happening and the probability of the same event NOT happening always adds up to 1'.
Another way to look at the Monty Hall problem is this: someone who sticks with their original choice no matter what will win just 1/3 of the time. Someone who switches will necessarily win when the sticker loses, and vice-versa. They never both win, or both lose. So the switcher must win 2/3 of the time because the sticker loses 2/3 of the time.
That sums it up
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