#21




Plus whether .999.... = 1. I think Monty Hall might be the most famous of the bunch though, so, yeah, probably that.

#22




I do (think I) remember being confused for a moment when I first heard the Monty Hall problem, but now the right answer is so obvious and intuitive that I can't remember why, and have trouble seeing why it's not obvious to everybody.
The one about the plane on the conveyor belt only causes arguments because the system is badly defined in the basic description, and people make different assumptions. (Apparently some assumptions are obvious to some people, and other incompatible ones to others). If you define the system to make the assumptions that you're making explicit, then the answer isn't hard  you just get a different one depending on the assumptions you choose. The arguments seem to be between people who don't recognise the other side's assumptions, or can't acknowledge their own assumptions. (eta) And the .9999... = 1 arguments are between people who understand maths, and people who don't but want to join in anyway. 
#23




I can gain 2 lbs in one day by eating a 2 oz piece of cake. Doesn't sound true but I've proven it many times.

#24




Quote:

#25




The setup: you choose from three boxes, knowing only one has a good prize. The host then reveals one unchosen box, showing it is a dud, and then gives you the choice to keep the box you have or take the one that is left.
Many people feel that once one dud is eliminated, that means the remaining boxes have a 5050 chance, but no. You see, there was a 2/3 chance that you initially selected a dud. The host is only ever going to reveal a dud in the next step, and there is always going to be a dud among the two unselected boxes. So the odds remain 2/3 that you initially selected a dud, and therefore you will win twice as often by switching when given the chance. 
#26




Twice as often?

#27




yes, 1/3 of the time you will lose by switching, and 2/3 of the time you win by switching. So you win twice as often by switching. (that is, if the game is replayed numerous times so that odds can manifest themselves. When you play it only once of course, it is simply that your odds are twice as good when you switch).
Hmm, now what if the two 'dud' boxes each had the different quantum states of Schrodinger's Cat, and when the host reveals the box in step two, it is always a cat, alive or dead? 
#28




Another math problem that people often get wrong is this. A certain disease strikes 1 out of 1000 people, totally at random. There's a test for the disease that's 95% accurate. If you have the disease, there's a 95% chance the test will say positive, and a 5% chance it says negative, and if you don't have the disease there's a 95% chance the test says negative, and a 5% chance it says positive. You take the test, and it comes back positive. What are the chances you have the disease?
Answer: About 2%. 
#29




In case anyone is wondering, testing 1,000 people would produce 51 positive tests, 50 false positives and 1 true positive. So if you tested positive, there is a 1/51 chance you are the true positive test.

#30




I once had a long argument with my dad about that one, Steve. I think Ben Goldacre (Bad Science guy) used it to illustrate a statistical point in one of his articles, and my dad was convinced he was wrong  which he thought was a shame because he also likes Ben Goldacre. Usually my dad is good at this stuff, but in this case he just couldn't see it.
I wasn't able to convince him of the right answer but I thought of a good way to do so later  although I've not tried it on him because I didn't have an opportunity for ages and I'm sure he's forgotten the conversation by now. Instead of a test for a disease, imagine it's a unicorn detector  and simplify the odds a bit. If you point it at a real live unicorn, it's 100% accurate and identifies it every time. But if you point it at any other object, it's only 99% accurate and gives a false positive 1% of the time. If you point it at a random object and it tells you the object is a unicorn, what's the chance that it actually is a unicorn? (Strictly speaking there's not enough information in my version to answer, but I'm assuming that people will see the point that real live unicorns don't exist. No linking to photos of rhinos or surgicallyaltered goats; those wouldn't fool my device.) 
#31




It's not quite that simple since there's a 5% chance it will miss the true positive  I think that's a factor so that the actual answer comes out to exactly 2% though. (I can't be arsed to do the sums now but I've done them before...)

#32




Love it! Plus, you get your potassium for the day.

#33




True things that sound false  Southern Moon
When I was in New Zealand, the full moon rose  and it was upside down!!! And there was a rabbit in it, not a face. But the bathwater still went down the plughole at random.

#34




Yeah, I sometimes point out that even if the disease is eradicated, we'd expect 5% of the population to test positive, which shows that you need to take the initial rate into consideration.
By the way, I'm not sure if it's exactly 2%, but why I tried Bayes formula I got 1.6% which is too low, so I'm screwing something up. 
#35




Quote:
If 20% of people have the disease  200 out of 1000, then there will be 190 true positives, and 10 false negatives, and of the 800 who don't have it, there will be 760 true negatives, and 40 false positives. That adds up to 230 positives (40 of them, or 17% false), and 770 negatives (10 of them false). If it comes back positive, 190 of the 230 positives are true, so the chance of having the disease is about 83%. I could go on, but suffice to say that this varies nonlinearly with the disease rate. I put it into a spreadsheet and for disease rates from 1% to 20%, there's a curve from 16% to 83% likelihood that a positive result means you really have the disease. As the disease rate goes smaller and smaller, the ratio of that "successful test" to "disease rate" approaches 19:1, or the success rate of the tests. It's really quite interesting to see in a spreadsheet. Suppose the disease rate is .0001%, so in 100 million people, you have 100 people with the disease. With the 95/5 accuracies on the test, you'll catch 95 out of the 100 people with the disease, but you will also get a whopping 4,999,995 false positives. The chance of a true positive, of all positives, is 95/4,999,995 or about 0.0019%, or 19 times the disease rate. 
#36




Quote:
The contestant made a random choice. The game show host is not making a random choice.  though I'm not sure why some chunk of my head seems to think that this provides more information than just the information that the door the host chose wasn't a winner. 
#37




Quote:
Obviously in reality, the actual disease rate is a massive factor in how useful the test is  but also, less obviously, we don't know what the actual disease rate is, because the test gives false positives... So you can work out the actual disease rate but it's complicated. Generally speaking the subtleties and complications get lost, though, and you do see similar reasoning used in reality to make apparently statistical points that aren't true, or aren't necessarily true. 
#38




I assume the tests which have high numbers of false positives, are for rare diseases. The clinical definition of "rare" vary, from 1/1,000 right up to 1/200,000, but the chances of a positive indicating true is always going to tend towards the accuracy of the test.
For example, multiple sources feed the wikipedia entry on false positives for AIDS/HIV  0.0004 to 0.0007, and a false negative of 0.003. Let's call that false positive at 0.0005 and assume an infection rate of 0.6% Running the numbers, the chance of a true positive is 92.3%, or 154 times the infection rate. Now that 0.6% infection rate does not make this a "rare" disease  if it were rare (say 1 in 1000 to 1 in 100,000) the true positive rate drops as the false positives climb. 
#39




I don't understand your computations. Especially this part.

#40




*sigh* Microsoft Excel works very well for this.
Using the above rates (true positive 99.7%, true negative 99.95%), an infection rate of 0.6%, and a population of 1 million, here's what I get. 1,000,000 total 994,000 healthy and 6,000 infected 5,982 true positives and 18 false negatives 993,503 true negatives and 497 false positives So 5,982 true positives out of 6,479 total positives or 92.329% chance that a positive is true. The ratio of 92.329% over 0.6% is 154  when the disease is very rare, this tends to an upper bound of the accuracy of the test, which in this case is 1:1990 Last edited by Hero_Mike; 08 February 2014 at 02:13 AM. Reason: All values rounded to zero decimal places per standard rules in Excel. 
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