'basic' probability

so, following through with my previous post of movies i had wanted to watch, and hadn't got around to 'til recently, i watched a movie today which i'd taken from a friend (N-dizzle rizzle, we'll call him, as he's related to H-dawg if only in the want to be black, as well) many eons ago. in any case, the storyline was wayyyyyyyy familiar kinda like deja vu watching it, so i guess i might have watched it at some point and just not remembered the title. it's an interesting piece titled '21' starring kevin spacey.

now, before i can attempt to launch myself into a repetitive entry similar to my previous one, i'd like to divert the course of my thought instead to something about the movie in particular that i found interesting (that, and the fact that i've not watched any other movie, which makes it slightly difficult to write a review).

as those who have watched the movie would guess (especially after seeing the title of this entry) it is about the probability-related problem posed by spacey in his lecture to the MIT students, as follows:

Suppose you are in a game show where you are presented with 3 doors; behind 2 doors are goats and behind one is a car. now suppose you have chosen door A, and the game show host reveals to you, after you have chosen, that behind door C is a goat. would it be in your best interests to remain with you choice of door A, or would it be advantageous to instead change to door B, given the option to do either?

knowing that this would be a trick question, i paused the movie before i could hear the answer, and began to ponder what i would do. for those interested in doing this yourself, take a time out and think of your solution before continuing. it is, for the mathematically inclined, as you have guessed, the classical Monty Hall problem (which i admit i had discussed during my first year of university, with a mathmo friend). at the time i think he presented me with the scenario word for word, and i did not agree with him, but as we'll proceed you might agree that it's just because of the phrasing of the question. rather, it's all in the wikipedia page dedicated to the Monty Hall problem (as linked above) so go read that if you've done your maths, or even if you haven't and just want the spoilers.

in any case, i won't bore you with the maths readily available from the linked page, but instead give you the conclusion of what i found (and in fact, justifies my answer from those many years ago). given no extra information, you would readily assume that the host has no obligation to show you a door behind which is a goat, and offer you the choice of changing your door A. if this were the case, then by all means, it's just a fresh random choice: it's 50-50.

if the question was (more accurately) re-worded:

Suppose you’re on a game show and you’re given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?

and from this wordy question, it is then easy to draw up a probability tree (or even doing the standard 'given' calculations A|B) and come up with the answer given in the movie that 'it is better to change your choice'. aside from the (rightly deserved, i think!) bragging-rights you have earned if you got the answer wrong (especially if you came up with the alternative answers associated with the myriad of possible interpretations of the question) you can now watch the movie with family/friends and at that precise point in the show, point out that the professor was giving insufficient information, with which there was no way the brilliant student could have come up with the perfect answer (like he did) without prior knowledge of the answer, or at least some understanding of the scenario! how awesome is that. unless you're in a crowd of 'jock-type' people, in which being perceived as the nerd would not be desirable.

ANYWAY. fun stuff, i spent the better part of an evening reading the Monty Hall problem and was thoroughly immersed in it such that i am now behind on what i should be doing. i hope the same fate befalls you all.