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1. ## Probablity puzzle

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. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

(assuming, of course, that you would rather have the car than one of the goats!).

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Slowly, Slowly catchee monkey...

2. ## Re: Probablity puzzle

I don't know the logic behind this (or probability factors) without thinking too much for a Wednesday when half the company is out, but I remember reading somewhere that your chances don't increase by changing your choice... Read it in something by Marilyn Vos Savant or whatever her name is who's in the Sunday papers and supposedly the Guiness Book of World Records for the highest IQ... (Even though she doesn't think the tests are valid! )

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"I want the fairy tale..."

3. ## Re: Probablity puzzle

I can't remember all the logic behind it (college was a long time ago), but you are better off switching doors.

I just found a web site that explains the theory. http://www.stat.sc.edu/~west/javahtm...MakeaDeal.html
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[This message has been edited by skootz (edited 11-21-2001).]

4. ## Re: Probablity puzzle

I'm going to go with no - changing doesn't increase or decrease your chances.

If you know that 1 of 3 doors has a car, the other 2 have goats, one of the goat doors is opened for you leaving 2 doors. Therefore, there's a 50/50 chance of getting the car!

If could argue that they wouldn't have bothered trying to change your mind if you had picked the goats. In that instance, you should switch. But a change from the normal way of doing things would raise alarms. So they always ask, so you can't use that argument!

So I'd say no, it doesn't matter, you have a 50/50 chance, either way.

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Hi Skootz
I went to that site - it seems to prove the theory that changing does increase your chances. But being a true QAer, I would have to question it because of the assumption that the applet has been programmed to specification or not. Was it programmed to reflect reality or the theory?

Skeptical to the end....

[This message has been edited by digits71 (edited 11-21-2001).]

5. ## Re: Probablity puzzle

Definitely switch doors. Each of the three options has a 1/3 probability of occurring. The danger is thinking that once you have one door out of the way, your remaining chances are 1/2. That is not the case.

You can make a matrix table of this to see how it works. You basically end up with a [6] to [3] probability matrix.

Remember that if you have a 1/3 chance of picking the correct door this means that since the probabilities must add up to one you also have a 2/3 chance of not picking the correct door. In other words, you are more likely not to win than you are to win.

So imagine that you open a door and see that there is a goat behind it. Obviously now the car is more likely to be behind a door other than the one you just opened up. Those two other doors have a greater probability, in other words, of concealing the car - and yet one still has a goat behind it. So you switch - because the remaining door now has a 2/3 chance of concealing the car. (Remember that your first choice still has a 1/3 probability of being the correct door, so the additional 2/3 probability must be somewhere else. Since you know that one of the two doors that previously shared the 2/3 probability does not hide the car, you should switch to the other door, which still has a 2/3 chance of concealing the car.)

[This message has been edited by JeffNyman (edited 11-21-2001).]

6. ## Re: Probablity puzzle

I don't remember the exact specifics of how to derive the solution...but the relationship of 2/3 failure during your original selection and the 1/2 failure during your secondary selection has something to do with it.

2/3 * 1/2 = 1/3. That's the equation I'm looking for...I just need to figure out how to work backwards to create the correct solution.

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Jason Trebilcock
Cyberentomological Detection, Prevention, and Eradication Specialist
Wells Fargo

7. ## Re: Probablity puzzle

Your skepticism is flawed, digits. As long as the applet doesn't show the door with the prize and randomizes the location of the prize(both of which it appears to do), then the applet shows BOTH the reality and the theory.

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[This message has been edited by skootz (edited 11-21-2001).]

8. ## Re: Probablity puzzle

Eventually I'd agree with you all, but I'm having a stubborn day.

I had this discussion with someone once over your chances of having a filly (female) or colt (male) when you bred your horse. They said that you have more of a chance of having a female if the previous pregnancy resulted in a male.

This isn't true because you have a 50:50 chance of getting either every time. However, if you begin to take into account of how *likely* is it that you'd have another filly if the previous 4 pregnancies resulted in fillies - then your odds would change, simply because the odds of having 5 of the same sex in a row are less! So there would be an equation somewhere taking the statistics into account.

That being said, my old mare had 2 colts and 2 fillies.

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9. ## Re: Probablity puzzle

Everyone - consider the solution by analogy. Say you can pick one lottery ticket among one hundred tickets where the winning ticket was among the one hundred. It is quite clear that the probability that the winning ticket is among the other 99 is 99/100, so that if all but one of those 99 tickets were revealed to be losing tickets, it would be in your best interest to switch, as the winning ticket was (and still is) among the other 99.

Or consider the same puzzle but say there are 1,000 doors. That means you have a 1/1,000 chance of pikcing the correct door. So if you open up 998 doors and all have goats behind them then the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Again, you are betting off switching.

This is not analogous, however, to a coin-flipping situation where each flip of the coin is independent and, thus, the chance of heads coming up is 1/2 each and every time.

So - finally, consider your three possibilities:

(1.) The game player first chooses the door with the car behind it. The player is then shown either door A or door B, which reveals a goat. If the player changes the choice of doors, then he loses. If he stays with the original choice, he wins.

(2.) The game player first chooses door A. The player is then shown door B, which has a goat behind it. If the player then switches to the remaining door, he wins the car. Otherwise, he loses.

(3.) The game player first chooses door B. The player is then is shown door A, which has a goat behind it. If the player then switches to the remaining door, he wins the car. Otherwise, he loses.

Each of the above three options has a 1/3 probability of occurring, as I said before, because the player is equally likely to begin by choosing any one of the three doors. There is no condition on that. In two of the above options, the player wins the car if he decides to switch doors; in only one of the options does he win if he does not switch doors. When he switches, he wins the car twice (the number of favorable outcomes) out of three possible options. Thus the probability of winning the car is 2/3 if he switches doors, which means that he should always switch doors.

(Note, however, that it is not a guarantee: just a statement of probability.)

[This message has been edited by JeffNyman (edited 11-21-2001).]

10. ## Re: Probablity puzzle

What are the exact rules of this game?

[This message has been edited by Yury (edited 11-22-2001).]

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