Do you know any mathematical "tricks" like variable change in the movie "21"?

Question

hi,
i just watched the movie "21" about the blackjack players, and there was a scene where the lecturer talks about the concept of variable change, where a game show host tells you there is a sports car behind one of 3 doors, but knows which doors just have goats (lol) behind them - after the host eliminates 1 of the goat doors he gives you the option to change your choice of doors, and if you do it gives you a 66% chance of winning the car rather than the 50% you might instinctively expect.

i thought this was cool as hell and wondered if anyone knew any other stuff like this, any other little "glitches in the matrix"??

please explain them if you do.
thanks.

Answer 1

Hrmmm... you're just asking if anything else in math doesn't fit "natural" intuition?



Sure, Banach-Tarski says that if we had a good enough razor, we could cut a sphere into a finite number of pieces, rotate and reassemble, and after you're done you'd have two copies of the original sphere.



Or the well-ordering theorem, but that's more theoretical.



Or my favorite topological theorem: You can't comb a hairy ball flat.

Answer 2

It's actually very simple. If there was no host, and only two doors left, of course, the new odds would be 50/50 (like in a coin toss, the previous flip has zero effect on the next flip- even if it came up heads 20 times, the odds are still 50/50). I don't care about 33% and 66% and how many doors there were. If at any point in time, there are only 2 doors left and you pick 1, your odds all day long are 50/50 IF THERE ARE NO OTHER VARIABLES. In this specific case, there is one additional and highly significant variable at play- THE HOST. Predicated upon the assumption that the host knows exactly where the car is, there is a statistically significant probability that it would be in the hosts best interest to have the player switch. Knowing that the player has the winning hand, the host is therefor motivated to have him change doors (change his "original decision.") If his "original decision" was incorrect, by contrast, it would be in the hosts interests to allow him to keep his original choice of doors. In this case, he simply offers him a choice that statistical research has proven, the player is likely to accept and will increase the odds of the host. Gambling is very much a science and in the long run, the house almost always wins. It's like in a coin toss, if the other person chooses heads and you peak and see it's heads, you might say, "are you sure?" It's really not rocket science but an interesting concept to think about on a rainy day. It's also a nice way to set up the movie as it provides contrast and irony to the plot but I digress- getting a bit off topic!

Answer 3

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