The answer to 1 is 2/3 or 66.67% for the closed door you didn't pick and therefore you should switch. This can be better seen if you focus on one thing and expand the example to 1000 doors.
So now there is a car behind 1 door and 999 goats. Think hard about the fact that the host knows where the goats/cars are and he will always reveal a goat, always.
You pick your door, say number 314. The host then opens 998 goats in a row and leaves door 769. Do you think you should switch?
I sure hope you do
Because the opening of the door is random in question 2, the chances are 50/50 now and it doesn't matter if you switch.
You could be correct that my phrasing of the question was wrong, hence my disclaimer at the end. The point I was trying to make was that you couldn't really be sacked on the strength of a positive result to that test as its actually likely to be a false positive than a real positive! You're looking at 100% accuracy before you trust tests for rare things!
Originally Posted by Andy
But by saying "the person tested positive" you're limiting the field down to a subset of the population which may not have the same percentage chance of taking the drug as the population as a whole.
Originally Posted by Andy
For instance, you have 100 people. 10 of them are doctors. 50% of them, including 9 of the doctors have degrees (just don't go to the doctor without a degree!).
So, taking a random person, he has a 50% chance of having a degree.
However, if you are then told he is a doctor (reducing to a subset).
Would you suggest the chance of him having a degree is still 50%? Or is it now 90% (given that 9 out of the 10 doctors have degrees)?
Or, even simpler, you have two cars in front of you on a game show, a red one and a blue one. You know if you pick the correct one you get to keep it, whereas if you pick the wrong one, you get nothing.
The chance of each one being right is 50%.
You are now told that the correct car is blue (you assume he is telling the truth - it's not a tricky game show).
Given that the "general population" of cars is still two, one of which is the winning car, is the chance of each one being right still 50%?
Take 100000 people
Originally Posted by AmeliesDad
On average, 500 people will take the drug.
If they all take the test, which is 99% effective:
495 will be correctly positive
5 will be falsely negative
98505 will be correctly negative
995 will be falsely positive
Given that your random person has a positive result, there are 1490 positive results, only 495 of which are correct, giving a (roughly) 33% chance of the person being a user.
Where is SuperDash didnt he just finish a degree in Statistics?
Have a plan and stick to it
For those interested in this sort of thing Peter Webb has an old site Probability Theory discussing various popular problems like the ones mentioned and some football ones too, Anatomy of a soccer match. Interesting read.
Answer is on the site, it was certainly less that I'd guess.
"How many people should be gathered in a room together before it is more likely than not that two of them share the same birthday?"
I'm trying to get my head around this explanation and just can't make it to make sense. A losing door will be revealed to you regardless of you picking a winning or a losing door so I just can't seem to figure it out how that can influence your chances.
Originally Posted by slim
My work here is done
Originally Posted by dulence
They all make your head hurt, it seems humans aren't very good at probabilities. Even top mathemeticians get these sorts of questions wrong, and even argue against them for a long time until they work out a proof that satisfies them. The other problem of course is that English isn't a very exact language. That's its beauty, and often in these cases its downfall.
Nothing influences your chances of having picked a winning door. What does influence your decision to switch or not at the end is that the host knows where the car is located and will never open that door.
So when you start you have a 1 in 1000 chance of picking the correct door. Therefore there is a 999 in 1000 (99.9%) chance that the car is behind one of the closed doors. Now the host will open 998 doors that all have goats behind them. Do you think it is 50/50 whether the car is behind your door or the unopened door? I hope you don't! The key is that the host knows where the car is and will only open goat doors.
If the host had no idea where the car was and randomly opened doors, it would be a different story.
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