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# Probability Problem. Really Interesting....

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John was a great thief. One fine day, when he was stealing jewels from the shop, he was caught red handed. So, he was produced before the king. On seeing him the king remembers that he was the one who saved his life from an accident a month before. But king has to give him a punishment too. So king himself took him to a prison. They arrived at a hall. There were 3 doors in the hall. King said, "Of these 3 doors 2 doors will lead to life-long prison and only one door leads to the way out for freedom. You saved me from my life. And now, You committed a sin. Anyway I can help you to some extent. You choose one of the 3 doors. I'll spot a wrong door from the rest 2. Then you can again choose the door you want and open it and get into it.". On earing this, John was happy that there's a chance to get himself free. He randomly chose a door. And from the remaining doors the King showed a door and said that that was the wrong one. Now John was confused which door to choose? 1. the door which he already had chosen? or 2. the door which is not chosen? Because this is his final choice, he's a bit confused. Now the question is choosing which of the two doors has a higher probability of escaping from imprisonment ?

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I believe I have an answer to this, and the solution is in the hidden section below. Its reasons like this that I'm glad I picked A Level Mechanics rather than Statistics

Hidden

If he keeps his current door he has a 33% chance of freedom, while switching would get him a 67% chance of freedom. There are 3 possibilities:

He has already picked the door to freedom, so switching puts him in prison.

He has picked the first door to prison, and the other door to prison is revealed, so switching would give him freedom.

He picks the second door to prison, the first door to prison is revealed, and switching leads to freedom.

Therefore, switching to the door he has not already chosen gives a 2/3 chance of going free.

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I've heard this problem before. When he chose the door the first time, the odds were 1/3 that he was right. Then one of the wrong doors is revealed, and the probability would seemingly be 1/2. But I think for some reason you should go with the other door because the probability that door is right is really 1/2 and the other was just 1/3? I can't remember, it's something similar to that. It's very interesting to look at and try to understand.Edit: Nevermind my reasoning, rvalkass has it correct.

Edited by husker (see edit history)

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@rvalkassGreat you found the answer very soon.

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Funny enough I did finish a course on statistics, but if you were to look to at the question a bit longer and not at rval correct answer, it would be obvious as to why his number is correct. Since the odds are lower with more choices they would slightly increase as more choices are dropped from the list. Then when the king chose a door those odds would only increase, but heres the thing the thif could be that got the wrong door as well so saying what rval says could be partially right. The number is correct but that number could swing both ways as to which door is the correct one.

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