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Probability puzzle

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louisg
Super Contributor
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 No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
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39 REPLIES 39
louisg
Super Contributor
That should read door No.1, No.2 and No.3
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THRESHOLD
Super Contributor
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Not applicable
No ... why would it? ... still 50/50
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THRESHOLD
Super Contributor
Actually the question wasn't framed correctly. The host must disclose the losing door and offer the switch thereafter. It has to do with the available outcomes of a switch once the original decision is made. If the first choice was good then a switch is bad. If the first choice was bad then either of the two switches are good - so a 2/3 improvement at the margin. A switch must, of course, only be made on a rational basis.
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Rams
Super Contributor
yes, agree. This analogy came from a movie about card counting.But what an odd combination, CARS and GOATS!
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G_V_V
Super Contributor
If you get it wrong on that game show you will have a greater chance on the second game show.
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THRESHOLD
Super Contributor
No. Each event is discreet and independant.
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THRESHOLD
Super Contributor
It is the fact that the events were no longer truly independent that skewed the outcome in the previous scenario.
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richardw
Super Contributor
Here's a discussion of the puzzle in general:
http://en.wikipedia.org/wiki/Monty_Hall_problem

Interestingly, Marilyn vos Savant (mentioned in the article) is quite an amazing character herself. She has/had an IQ of 185, which at the time got her a Guinness World Record for high IQ. This particular question got thousands of well-educated people arguing about her solution, but she was proved right. I still find the result unintuitive though.
http://en.wikipedia.org/wiki/Marilyn_vos_Savant
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G_V_V
Super Contributor
I don't believe that when you spin a coin there is a fifty fifty chance that it will fall on heads or tails to the previous spin. Only the first spin in your life has a fifty fifty chance what happens after that depends on your previous spin. If you believe each spin has a fifty fifty chance then you would spin heads and tails repeatedly forever to make your analogy true that each spin has a fifty fifty chance. The mere fact that there is a formula that determines pressisly on average how the sets of 1,2,3,4....heads and tails fall in a row proves that it is not a fifty fifty chance. T=2((2(to the power of n+1))-(n+2)) Where "T" stands for the number of times a coin is tossed; and "n" stands for the level of x number combinations tossed. For example, to determine how many tosses, it will take to reach 7 heads and 7 tails in a row. The answer is 494. The "n+1" equals, 7+1=8. Two to the power of 8=256. The "n+2" equals, 7+2=9, subtract it from 256=247, times it by 2=494.
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THRESHOLD
Super Contributor
Butterfly effect - not mainstream and unsupported. Or you subscribe to the notion that there is a predetermined finite numer of events in a sequence. Otherwise you are saying that the field of statistics is faulty; and we are not qualified to have that arguement. It would be pure folly.
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G_V_V
Super Contributor
Yes I subscribe to the notion that there is a predetermined finite numer of events in a sequence and if the number of events breachers the sequence number of events become part of the next sequence. It is like building a pyramid with the sequence of one's at the bottom half the sequence of two's than one's above the one's, half the sequence of three's than two's above the the two's and so forth until the highest sequence is at the top. The pyramid is formed at the sides higher and higher which the lower sequence two times more than the higher sequence.
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louisg
Super Contributor
Logic dictates that one should ALWAYS switch. You would have a 2 in 3 chance when you switch. Or put another way, you only have a 1 in 3 chance if you DON'T switch.(The puzzle comes from an American game show called "Let's make a deal")
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BlueDolphin
Frequent Contributor
This quote from Wikipedia I liked most - "Interestingly, pigeons make mistakes and learn from mistakes, and experiments, Herbranson and Schroeder, 2010, show that they rapidly learn to always switch, unlike humans."
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louisg
Super Contributor
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Kabeer
New Contributor
Don't know if this has been said yet, but.. Imagine there were 100 doors, u choose 1, 98 doors eliminated, do u swap? After the elimination its 50/50 but YES U MUST SWAP. I would swap with 3 doors and 1 gets eliminated.
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louisg
Super Contributor
After the elimination it is not 50/50. The odds are set before you choose a door and not after the elimination. The odds are always 1/99(with 100 doors) before you choose.ie.you have a 1% chance of choosing the correct door.
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