Maybe I didn't make the directions clear enoughGood job Phil, or have you heard this one already.
Alright let me try to reexplain the contest.
You pick your door. There are two doors left now that you didn't pick. Out of the two remaining doors
that you didn't pick, the host opens one and removes it from the game (that door never has the car behind it). Now there are two closed doors left, one is the door you originally picked and the other is the door that was left after removing one of the doors that didn't have the car behind it. Now either the door you originally picked has the car behind it or the other door. Do you stick with your original choice or do you switch to the other door.
Door   
Well from the beginning, each door has an equal probability of having the car. Therefore if you were going to pick a door and then stick with it no matter what, you would have a 1/3 chance of winning. But if you pick a door, there is a combined chance of 2/3 that the car is in one of the other two doors. So think about it like this, since the host removes one of the remaining two doors without the car in it, the situation basically is equivalent to saying that the host is allowing you to have car if it is in either of the remaining two doors if you decide to switch
. Its almost like allowing you to pick both of the remaining two doors if you decide to switch.
I like this puzzle because its weird and it shows that even though the finial decision is between two items (you would think it is 50:50), there is not an equal probability between the two choices. Even from a pure probabilistic stance, the previous history of where the choices came from comes into play. Haha, I don't even have a minor in math (but I did take calc 5
), so some of these terms that I used I might have made up, but bear with me. The fallacy is like saying it will either rain or it will not rain so theferfore there is a 50:50 chance. We know that it doesn't rain half the time, so there must be more to the equation.
Probability that we are used to is the type that you almost have to assume there are equal probability between choices. For instance if you are throwing rocks onto a 4x4 grid with equal area blocks and a rock has to land somewhere on the grid, the total possibilities are 16 so we have to assume that each of the possibilities are equiprobably in order to say that there is a 1/16 chance of landing on each spot. That is the basis probability. But what if your style of throw favors you to land in a particular range of squares. In this case the probabily between the possibilites is not equal so you can't just use pure probability alone. Its almost like circular logic that you can only use probability if there is equal probability....
Something weird that I like think about is that if you were were to try to predict the probability (not from a probability standpoint) of something occuring at a future instance, and if you were to incorporate every single possible variable into your analysis of the probability, the more variables you include, the closer you would get to 0% chance or 100% chance because in that future instance, what you are deriving the probability for can either happen or it won't happen. This is to say that you believe in determis, which I sorta do, and you don't believe in anything from quantum mechanics or natural randomness.
And to get completely off topic and talk about determism
If you believe in physics, you know that everything has a prescribed way of happening (whether we have the capabilities to understand the governing equations or the ability to account for the vast number of variables), and given a large enough set of initial conditions, there can be only answer to what will happen.
Well what about quantum mechanics, and the seemingly random occurance of stuff. Well lets say that things are random at the quantum level... Do you believe in free will?? I do, but if you think about it like this (what im about to say) then it becomes confusing.
At any given point in time, there are a vast number of conditions that occur. At the next instant (bear with me, whether you believe in quantum time units or not, you can get the gyst of what I'm trying to say) what happens has to be determined by the
1.) conditions from the previous instant and
2.) random stuff the goes on
So from a persons point of view, he only "has" control of those conditions that were passed over from the previous instant. Well if you make an infinite regression you back in time from instant to instant, you can see that the only control of person has over his whole life is the conditions that occured at the "beggining of time", which when you think about it, a person doesn't even have control of that (he wasn't even born yet). So nothing that happens to or will happen to a person is in any way controlled by the person's free will, but rather enivitability.
Well, if you followed that, you can see how its confusing. Its like a mathematical proof, sometimes in a math proof you can follow each step with clarity and each step is justified but what all the steps lead to is an end which is not at all clear and hard to accept (hence the usefulness of math proofs). So are we suppose to look at this proof and say well I have no control over anything and lose our grip on reality. No, its not something you can or should really accept other than just being interesting to note.