November 02, 2006

• There is a 59% chance of what that the taxi was green? That the taxi the person saw was green or that the person would see a green cab get in a wreck? It isn't clear to me what the question is about. Are you more likely to see a green cab in an accident or are you more likely to be correct about the color?
• Condensed version: 1 taxi commits hit and run. There are only two colours of taxis in town: 85% are green, 15% are blue. The sole witness of the accident is 80% reliable -- 20% of the time that witness will say a blue taxi is green, or vice versa. Witness says taxi was blue. What's more likely to be the colour of the hit and run taxi?
• There are four total possible scenarios: A. It was actually blue, and you identified it correctly as blue; 0.15 * 0.80 = 0.12 B. It was actually blue, but you misidentified it as green; 0.15 * 0.20 = 0.03 C. It was actually green, and you identified it correctly as green; 0.85 * 0.80 = 0.68 D. It was actually green, but you misidentified it as blue; 0.85 * 0.20 = 0.17 0.12 + 0.03 + 0.68 + 0.17 = 1.00 Given that you identified it as blue, that means it had to have been A or D. The total probability for (A or D) is (0.12 + 0.17 = 0.29) and from that you can see that D will happen (0.17 / 0.29 = 59%) of the time. You can almost always solve these probability problems correctly by just enumerating all possible outcomes and the probability of each, and then summing the ones you are interested in.
• Look at some extreme examples of this: if there were zero blue taxis in the town, but you identified the sideswiper as a blue taxi, the odds of the taxi actually being blue would still be zero regardless of your percent success rate at identification. On the other hand, if you were 100% successful at identifying the taxi's color, the ratio of the taxi colors wouldn't matter. In any intermediate case, the color of the taxi does depend on both your skill as a witness and the ratio of the town's taxi colors.
• And on another note, do you really believe that a book on probability would get something this fundamental wrong?
• Or alternatively:

  Whatever the likelihood that one colour or another is presented to you, you have an 80% chance of identifying the correct colour.

  But that is not the question you are answering. The question you are answering is "what actual color was the taxi?" And the answer to that will depend on both the ratio of green to blue taxis on the street and at how well you are at judging color at night. Because green taxis are much more common, it turns out to me more likely that you misidentified a green taxi as blue than you saw a rare blue taxi and identified it correctly.
• Bayes' Theorem.
• Look up bayesian statistics if you have anymore questions and you'll find all kinds of explanations for this calculation.
• But, but, but... Aren't there really just two possible scenarios: 1) The car had a colour and you correctly identified that colour: 80% 2) The car had a colour and you incorrectly identified it: 20%.
• That is true, but it's not the question that you are answering! The question is not whether you identified it correctly, it's what was the actual color of the taxi. Those are not the same question. Whether or not you identified it correctly is only half of the matter.
• This sounds like a job for renowned monkey-mathematician petebest. There's also a statistically significant chance that a taxi from a nearby town, where they have purple taxis, had dropped off a passenger in the town in question.
• if there were zero blue taxis in the town, but you identified the sideswiper as a blue taxi, the odds of the taxi actually being blue would still be zero regardless of your percent success rate at identification. Agreed. There's no problem regarding probability because all taxis are one colour. Success rates at distinguishing between various colours is irrelevant if there is only one colour. On the other hand, if you were 100% successful at identifying the taxi's color, the ratio of the taxi colors wouldn't matter. In this example, the skill of the witness operates independently of the ratio of taxis. 100% is 100%, regardless of how many taxis in town are blue or green. Similarly, I suppose, if the witness was always wrong -- 0% -- that would operate independently of the taxi colour ratio. So if at the extremes, witness skill is not dependant on the overall ratio, why does it necessarily have to be linked in cases of somewhat skilled witnesses, like those with 80% success rates?
• 85% of the time I fling black poop, 15% of the time I fling brown poop. 80% of the time the poop I fling hits the person I flung it at. What are the chances I am going to fling poop at you and run? /flings poop and runs.
• That is true, but it's not the question that you are answering! The question is not whether you identified it correctly, it's what was the actual color of the taxi. Those are not the same question. Whether or not you identified it correctly is only half of the matter. Whoa. My head hurts. Is there a way for the witness to identify the colour correctly other than stating the actual colour of the taxi?
• Identifying the color it isn't?
• Try reformulating the problem. Suppose you have a lottery ticket. The chances of it being a winning ticket are one-in-a-brazillion, i.e. vanishingly small. It doesn't matter what they actually are. Now suppose that your eyesight is not what it used to be and that you misread the numbers on the ticket one in twenty times that you look at it. Now, suppose that you look at your ticket and think you've won. Which is more likely, that you didn't actually win but instead misread the numbers, or that you did win *and* read the ticket correctly? Wouldn't you say it was much more likely that it was the former? See how this pits the chances of misreading against the chances of winning, and because the chances of winning are much smaller than the chances of misreading that it's almost certain that you misread a number? Now just transpose "chance of winning" with "car was blue" and "misreading" with "misidentifying the color of a car" and you've got the same basic thing.
• Monkey Flitter's example, while cowardly and stinky, is also quite illustrative: -- Likelihood of poop being brown: 85% -- Likelihood of poop of any colour hitting target: 80% But, but, but... In the taxi example: -- Likelihood of a random taxi being green: 85% -- Likelihood of a specific taxi that is identified as green actually being green: 80% So, in the monkey example, the two stats given are about different variables, colour of poop and likelihood of hitting me. But in the taxi example, the two stats are about the same variable: colour of the taxi. Doesn't this make a big difference to Baye's Theorem? I just had a look at the Wiki link, and it says that: "Pr(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B." In the monkey poop example, the colour of the poop and the hit rate are unrelated, so one can be "prior" of the other. But in the taxi example, can you say that one fact is truly prior of the other?
• Tsk. Perhaps you're looking for an argument, not an answer. If it's any consolation this kind of math is counterintuitive and frustrating for me, as well. May I recommend Cecil Adams's discussion of the Monty Hall problem? If you still don't agree, try this simple simulation.
• Applause to Rhomboid and the other Monkeys above who obviously have a firm grasp of statistics and probability (which, in my world is intelligent design: there is no design, it is just intelligence). I have always thought there was a mathematical beauty to the study of probability, something that I can see and feel but cannot describe in words. Get out of my brain, it is disorganized and dangerous.
• How about a magic trick instead? Dark taxi hits car. Bystander witnesses accident. Without memorizing the license plate or the 25-digit ID number on the "How's my driving" sticker, properly identify to police the actual vehicle without the aid of inspecting any taxi during daylight.
• The taxi was yellow. I have dealt with probability before with characters like Carnap and people like that, so I have a question. What happens to the probablity of someone identifying the correct car color become after the witness gives his answer, right or wrong? See the problem is that the person is equally likely to answer blue or green. Given that there is ever a chance of error, the answer always has a 50/50 chance of being blue or green. The number of cars are irrelevant to the equation. To use an example, with flipping a coin, it is commonly held that if you do it enough you come to a 50/50 result. Now if you toss a coin 100 times and you get 80 heads and 20 tails, people will argue that your sample wasn't big enough and results will eventually head to 50/50. We believe this because of the physical nature of the coin. But we cannot explain the cognitive reason for the 20% error. Knowing that we cannot explain that 20% we can only assume that they have an equal chance of saying blue as green, and each one has just as much chance as being wrong. "20%! well your sample size wasn't big enough." A seperate question would be "What is his chance of running into a blue vs green taxi?" But that does not give insight into what the witness perceived.
• At least two further issues should, in the real world, be considered: 1.the colour of the available light source and 2.whether the individual witness suffers from either red-green or blue-yellow 'colour blindness'. Coin flipping is slightly biased toward a coin flip's result being similar to its starting position.
• Glamajamma, seriously, wtf are you trying to say up there? Who knew mofi would have an fpp relevant to the midterm I am taking tomorrow? Go figure. On an unrelated note, hey Rhomboid, whatcha doing tomorrow afternoon at 1pm pacific time?
• Bees is right. Were the streetlamps those funny yellow-colored-light ones? Cause that'd make a green car look blue, or a blue car look green, or something. Also, are you drunk when you witness this hit and run? And if so, have you been drinking curacao, or creme de menthe? And assuming that you're drunk, what's the probability that you're more focused on getting home to take a pee than on accurately surveying your surroundings en route?
• Further: Is glama drunk? Or more to the point: Just how drunk is glama?
• Guys, check this out: there is no taxi.
• Ha!
• The chances of ever seeing a unicorn are 1 in a million. You claim to have seen a unicorn, and are known to have a 99.9% reliability rate concerning your ability to recall events. What are the chances that you actually saw a unicorn? Can you see how your reliability isn't the only factor? The likelihood of the event needs to be factored in, along with your reliability. Here's another example: You are applying for a new job at a company that has a requirement for prospective employees to take a drug test prior to employment. This drug test has a 95% reliability rate. Of all the people that apply for this job, 10% are drug users, the other 90% are drug free. Your test comes back positive. What are the chances that you are actually a drug user? There's a 21.1% likelihood you are a drug user.
• Here, the use of “prior” doesn’t imply “event A happens before event B” or vice versa or really any relationship between A and B. It’s more like this: Pr(A) is the probability of A happening. Period. Prior to us even thinking about B’s influence. In fact, we find Pr(A) by adding up all the possible ways (within the restrictions of the problem) that A can occur, regardless of whether or not B happened. But in the taxi example, the two stats are about the same variable: colour of the taxi. Actually there are two variables, actual color and perceived color. Independently, the actual color can be green or blue, and the perceived color can be green or blue. For the probability we want, the two events are “The taxi’s actual color is green” (Ag) and “I see a blue taxi” (Sb). We want the probability of the taxi being green, which, with no other information is 85% (this is the prior probability, Pr(Ag)). So already it’s pretty likely it was green. But we have the extra info that I saw a blue taxi, so that makes it less likely that the taxi is green (if you trust my vision at all), but by how much? To find out, we have to look at my performance in another hypothetical experiment. Out of 100 taxis, 15 blue and 85 green, I will see 12 blue taxis as blue and 17 green taxis as blue, or 29 total perceived blue taxis. I mistook 17 green taxis as blue, so among my blue sightings, I mistook a green taxi for blue 17/29, or 59%, of the time. This is the same as Pr(Ag | Sb) or the probability of the taxi actually being green restricted to those situations where I saw a blue taxi. That’s just a rewording of Rhomboid’s original enumerative method, which for me is the cleanest way of looking at it. Also looking at extreme examples as Mr. K and Rhomboid have done helps the intuition a lot. I tried and failed to find an intuitive way to think about the ratio Pr(Sb | Ag) / Pr(Sb) = 20/29, which in Bayes’ Theorem is multiplied by 85% to get what we want. Oh well, intuition sometimes just isn't there.
• Or you could just check the paint scratches on the car that was hit.
• *holds hands over ears* la la la la not listening....
• See also Dempster-Shafer theory for thought-provoking developments on (or away from) Bayesian probability. An interesting court case that relied on Bayesian analysis. There was a decent telly documentary on this a few years back.
• I am saying without any explination to why there is a 20% margin of error that has occurred among the sample that was apparently taken from a past sample, you cannot intuit the reason for error, so we can only assume that the percentage of error is a mean 50%, because it is just as likely for the error percentage to rise or fall. See the reason we assume the coin will balance to 50% head and 50% tails, because we have insight to the qualities of the coin, but we can't have insight into the reason for the 20% error, considering all things were equal in past test subjects. Without the insight we cannot intuit if we have EVER taken a large enough sample size, so we can only assume, given all possible answers he is 50% likely to get it right and 50% chance of getting it wrong.
• Is this the bathroom in India thread?
• Rhomboid's lottery ticket example is an excellent basis on which to further this conversation. With the lottery ticket there are three "balls in the air": 1. The winning number. 2. The actual number on the ticket held in the hand of the guy with sight problems. 3. The number the guy with sight problems observes. As Rhomboid points out, there is an extremely small likelihood that the actual number on the ticket is the same as the winning ticket. So that first variable -- the likelihood of variation between the winning number and the number on the ticket -- has to be taken into account. But what if we change the problem? Now let's say that the ticket the sight-challenged guy holds in his hand is the winning ticket. In other words, #1, the winning number, and #2, the number on the ticket, are the same. So, what are the odds that he is a reliable witness? How likely is it that he can properly observe that he has the winning ticket? You'd be barking up the wrong tree if you started off your analysis by saying -- "well, the odds of him holding the correct ticket are one in a zillion." You don't have to go there because the question now defines the likelihood of his holding the correct ticket as 1 in 1. 100%. The only variable left is the possible variation between the actual number in his hand, and what he will perceive that number to be. We know he's correct 19 times out of 20. So his odds of his figuring out he holds the correct ticket are 19/20 to the power of the number of digits on the ticket. Ok, back to the taxis. The three "balls in the air" in the taxi problem are: 1. The true colour of the taxi in the accident. 2. The actual colour of the taxi seen by the witness. 3. The colour perceived by the witness. What are the odds that #1 (the colour of the taxi in the accident) and #2 (the actual colour of the taxi seen by the witness) are the same? We know from the question that she saw the accident. She didn't see a random taxi that may or may not have been the one in the accident. She saw the specific, particular taxi that committed the hit and run. So there is no variation between #1 and #2 in the taxi problem. The odds of #1 and #2 being the same are 1 in 1, or 100%. If you assume a variation between #1 and #2 -- using the 85% blue, 15% green stat -- I believe you make a logical error. To put it another way, because we know she is oberserving the specific taxi in the accident, she is, metaphorically speaking, holding the winning lottery ticket in her hand. The only variable in the question is the variation between the actual colour of the taxi, and the colour she perceives. She right 80% of the time. Since we're only dealing with one taxi -- not the multiple digits on a lottery ticket -- the odds are a straight up 80% chance she's right. Or have I made a mistake somewhere?
• yes
• Torluath, bear in mind that you're trying to arrive at a probabilistic statement - one that represents the most likely answer. Take a somewhat different problem to consider why the proportion of taxis on the road is important: Let's say there are the same number of blue taxis and green taxis available, but on a given night there are more of one color on the road (e.g. on Monday, it's 85% green; on Friday it's 85% blue). Let's say the accident happens five times, and each time the witness identifies a blue taxi. One of these identifications is wrong, according to the tests of the witness's ability. Is it possible to establish which identification is most likely to be wrong without looking at the number of blue taxis and green taxis on the road on a given night?
• Has no one given any thought to the incidence of blue-green colourblindness among men contra among women?
• OK, Bees did.
• One of these identifications is wrong, according to the tests of the witness's ability. Wouldn't there be a 33% chance that the witness has it right?... 1st event -- 80% chance correct 2nd event -- .8 x .8 = 64% chance correct 3rd event -- .8 x .8 x .8 = 51% 4th event -- .8 x .8 x .8 x .8 = 41% 5th event -- .8 x .8 x .8 x .8 x .8 = 33%
• There's a 33% likelihood that the witness gets all five correct. As the events are not dependent on one another, the witness could be wrong each time or no times. However, if we asssume an 80% success rate, we expect that the witness is wrong once out of five times. This isn't the important part though. Given that the witness is wrong once, what is the best guess on which night the witness made the mistake?
• Ha ha, you LOSERS! It's a TRICK QUESTION! It's NIGHT and you CAN'T SEE ANYTHING! MORANS! I WAVE MY FURRY BITS at you in DISGUST!
• ALTERNATIVELY, the car was going SO FAST it created a DOPPLER SHIFT, and what APPEARS to be a BLUE CAR is ACTUALLY GREEN.
• MOREOVER, I am just a BRAIN IN A VAT, and there is NO CAR either GREEN OR BLUE, and you ALL EXIST within the CONFINES OF MY MIGHTY IMAGINATION. Now SERVICE ME, my MINIONS!
• *hands the capt. his meds*
• Roryk -- Agreed, when you shift around the total blue and green taxi ratio, the ratio of blue to green in the 80% correct bunch will shift. And, of course, the ratio of blue to green in the 20% error group will shift as well. But for each individual instance, the success rate still sits at 80%, whatever the actual colour of the taxi. The decision of the taxi dispatcher to send out a bunch of green taxis one night doesn't suddenly drop my reliability as a witness from 80% to 59%. Even on a night when greens greatly outnumber blue, I can still properly identify blue 80% of the time.
• Hey what if the guy gets the color right, but the local police fall under the 20% that get it wrong?
• Think of it this way... I see somebody steal candy from a baby. He's wearing a baseball cap and a gorilla suit. I'm a pretty observant guy. I can spot caps and gorilla suits equally well -- 90% of the time. 45% of the population of my city wear baseball caps. .00001% of the population of my city wear gorilla suits. At the trial, should you treat my evidence about the gorilla suit as less persuasive than the evidence about the baseball cap? Am I more likely to be wrong about the gorilla suit just because it's unusual?
• This is one of those mathematical ponderings that have little or no relevance to a real-world accident. Where there may be a host of variables - like the sobriety level/druggedness of the witness, general health of witness (does s/he have Alzheimer's or macular degenration? etc), colourblind status, type of light-source, etc. Statistics are irrelevant in dealing with an actual and particular event and individual situations. Statistics deal with averages and means, and aren't all that relevant when considering individual happenings. Freaky, unpredictable, unforeseen, and/or unusual things do happen. Statistics can only average such happenings into charts and tables and so on, but overall those have no particular pertinence or relevance to any one given real-life situation.
• By the great gods of synchronicity, Torluath, I am reading the exact same book! I love it, it's one of the few math/science books that are well-written as well as informative. Okay I got to throw my 2 cents in: We know from the question that she saw the accident. She didn't see a random taxi that may or may not have been the one in the accident. She saw the specific, particular taxi that committed the hit and run. So there is no variation between #1 and #2 in the taxi problem. The odds of #1 and #2 being the same are 1 in 1, or 100%. She saw the taxi BUT we already know she is only 80% accurate in her "seeing". So the odds of #1 and #2 are not the same. That is the crux of it.
• This is one of those mathematical ponderings that have little or no relevance to a real-world accident. You are high, bees. This kind of statistical analysis is extremely relevant to the real world. I gave two examples above. It not just a math trick, it's the way the real world operates. The fact that this is so counterintuitive to so many people has harmed society in so many different ways.
• That's a good question, StoryBored: Is she 80% sure she saw the hit and run taxi? Or is she 100% sure she saw it, but only 80% sure of what colour it is? I thought the latter. But the former would make much more sense. Wen righting, klaritee and persision r aul.
• Bingo! These questions always bake my brain and I end up with cranial hurt. She is 80% sure of the color. So even if there was an endless stream of green taxis, she will still misidentify some of them as blue. The problem formulation is unrealistic and doesn't help. Maybe a better example would be a doctor trying to make a diagnosis from a mammogram or something like that.
• I think it's clear that we're intended to consider it guaranteed that you really saw the taxi, and you're just not 100% sure of the color. Whether or not that's realistic is irrelevant. Imagine setting up an experiment where our guy watches a computer animation of the hit and run accident many, many times - 15% blue taxi animations and 85% green taxi animations. Which is going to happen more often...that he gets the color wrong, or that the color is blue? Since he kind of sucks at seeing blue stuff, it's going to be the former, so the probability you're looking for has to be more than 50%. Your gorilla suit example is misleading because intuitively, we expect someone to be extremely reliable in identifying whether or not another human being is wearing a gorilla suit, not 90%/10% as you suggest. You *are* more likely to be wrong about the gorilla suit because it's unusual. Seems counterintuitive, but in fact the part of your example that conflicts with our intuition is not the probability, but the amazingly bad job your witness does of identifying gorilla suits. He'd be thinking that 1 out of every 10 people he sees is wearing one! So to account for that, what if your witness was also tripping on acid at the time? Would you rely purely on his testimony and set the gorilla-suitless, cap-wearing suspect with a long history of candy thievery free, or would you think, maybe this isn't the most reliable witness? What if your witness also claimed that the thief was 100 feet tall, made of blueberries. and carrying an ipod? Which of those three observations are you going to take most seriously? In a trial, you'd generally want to throw out the extremely improbable claims.
• 100ft tall blueberry man sounds delicious.
• Oo, i should also mention my favourite sentence from the book so far. It's on page 88 right after a formidable looking description of the Weak Law of Large Numbers. It goes: "This formula, like all others, is a kind of bouillon cube, the result of intense progressive evaporation of a larger diffuse mix of thought. Like the cube, it can be very useful -- but it isn't intended to be consumed raw." Cool! While we're exercising brains here, I also have a question about the Law of Large Numbers. The Law states that for any given degree of accuracy, there is a finite number of observations necessary to achieve that degree of accuracy. The Law is expressed in mathematical form. But what were the underlying mathematical assumptions behind it? Because in a way, this Law seems to be a metaphysical statement.
• Imagine setting up an experiment where our guy watches a computer animation of the hit and run accident many, many times - 15% blue taxi animations and 85% green taxi animations. Which is going to happen more often...that he gets the color wrong, or that the color is blue? Since he kind of sucks at seeing blue stuff, it's going to be the former, so the probability you're looking for has to be more than 50%. Over many, many accident simulations, there will be a total universe of n accidents of which 15% will be blue and 85% will be green. And the 80-20 right-wrong ratio will sort itself out as Rhomboid described. But in each individual case, the odds are always 80-20. This is perfectly consistent with Rhomboid's numbers: A. It was actually blue, and you identified it correctly as blue; 0.15 * 0.80 = 0.12 B. It was actually blue, but you misidentified it as green; 0.15 * 0.20 = 0.03 C. It was actually green, and you identified it correctly as green; 0.85 * 0.80 = 0.68 D. It was actually green, but you misidentified it as blue; 0.85 * 0.20 = 0.17 0.12 + 0.03 + 0.68 + 0.17 = 1.00 Over many, many sightings, when you look in the 80% pile, you'll see a ratio of 68 green and 12 blue (17 to 3). But for any specific sighting -- whether it's blue or green -- the ratio of right to wrong answers will always be the same: 4 out of 5 times (80%). If the witness says a specific taxi is blue, Rhomboid's numbers say that in 12 out of 15 times (80%), that will be correct. If the witness says it's green, his numbers say that will be correct 68 out of 85 times (80%). Whatever the proportion of taxis on the road, you can count on the witness to correctly identify them 4 out of 5 times.
• Torluath I think you just need to acknowledge that you don't know jack shit about statistics. You're wrong.
• Torluath, it's not so much that you're wrong, it's just that you're wrong about the stated problem. It seems your intuition is generating a problem that isn't quite the same as the original stated problem. It's true that given, say, a blue taxi, the witness will correctly identify it as blue 4 out of 5 times. This is the probability of seeing a blue taxi given that it actually is a blue taxi. But in the problem we're trying to solve, we don't know the actual color of the taxi. All we're given is what the witness saw. We're trying to find the probability that the car is actually blue given that the witness saw blue. This is not the same question and not the same probability as before, though it may seem like it. I think it may be because, in the absence of any other information, we tend to fall back on the reliability of the witness. But even if we don't know the distribution of blue and green taxis in the town and assume it's equally likely to encounter either, the two probabilities are still not the same. It may be possible that if we make no assumptions about what color cars are in the town, and thus have an infinitude of possibilities, then these two probabilities might converge. I'm not sure though, I'll have to think about it. But I do know that in order for the two probabilities to be equal in the original problem, some weird shit has to happen. I think an understanding and an agreement can be made about our intuitive understandings of this problem, as contrived as it is. It's not like this is a religious argument (not yet, anyway).
• Help me out, Xerxexrex. I agree that... We're trying to find the probability that the car is actually blue given that the witness saw blue. But if it's given that the witness says "blue" I don't get how it can be among the possiblities that "C. it was green and you identified it as green". If it's a given that the witness says "blue", how can you assign a probability to the chance that the witness says "green"? It's a 100% certainty the witness said blue. And as for you, Rhomboid, I forgive you for your uncouth words.
• OK, I'm fine with the whole blue car green car thing, but now I'm confused about Mr Knickerbocker's drug test problem. If, as stated above, the drug test is 95% accurate and 10% of the prospective employees are drug users, wouldn't I be about 67.86% likely to be a drug use if my test comes back positive? Using the methodology above, I see 4 possible outcomes: User tests positive, user tests negative, clean person tests positive, clean person tests negative. So... User/positive = (.1)(.95) = .095 user/negative = (.1)(.05) = .005 clean/positive = (.9)(.05) = .045 clean/negative = (.9)(.95) = .855 If the test is positive, there are 2 possible outcomes: A or C. .095 + .045 = .14, Probability the person who tested positive is a user is .095 / .14 = .6786 What am I doing wrong???
• Xerxexrex said it well : "It seems your intuition is generating a problem that isn't quite the same as the original stated problem." The difference is as he describes it. It's subtle and non-trivial. The trouble is that it's hard to see "it" (the intended problem) without a mental flip.
• Torluath, would this help? Let's just keep the question the way it is except for one small change. The witness still has 80% reliability but now Blue Taxis are .001% and Green Taxis are 99.999%. For the witness to have 80% reliability over the long run, there will be 4 right guesses for every wrong guess. Over 100,000 accidents, the witness makes approximately 20,000 mistakes. Over 100,000 accidents, you would expect there will be 99,999 Green taxis and 1 Blue taxi. Your witness comes running to you during this long run of accidents and claims she saw a Blue taxi. Are you going to believe her when you know that 20,000 of her guesses are wrong and that roughly 99,999 Green taxis will be observed for a single blue one?
• Probability the person who tested positive is a user is .095 / .14 = .6786 What am I doing wrong??? Nothing as far as I can see. You did it right. I didn't catch it before, but I don't know where the 21.1% comes from.
• Now let's keep going. The witness has 80% reliability, but now the percentages are 1 Blue taxi for every 99,999,999 Greens. It is essentially a green taxi ocean with a single droplet of blue. In 100 million accidents, the witness will have observed roughly 20,000,000 false blue taxis. When the witness reports blue, you can be pretty sure she is wrong. She most definitely is not 80% right.
• jaypro22: You're right, it's 67%. I messed up my math. Good catch.
• ))) for StoryBored's reformulation > the Law of Large Numbers The Wolfram website (a fantastic resource) gives proofs for the weak and strong forms.
• I had a delicious steak dinner with my lovely wife last night, had a couple glasses of good red wine, took the dog out for a walk in the brisk November evening air, and the penny dropped. It's true that in any individual case the witness will be 80% likely to be correct. But the wrinkle is: exactly how many opportunities do you have to be correct? When the witness says "blue", there are only 2 possibilities: 1. Correct, the taxi is blue, or 2. Wrong, the taxi is really green. So, how big is the correct pile? How many yes's can there be? Out of every hundred taxis, we know 15 are blue, and with an 80% accuracy rate, that means the correct pile will be 12 taxis. How big is the wrong pile? How many no's can there be? If the witness always identifies 80% as correct, she'll identify 67 taxis as green. That leaves 17 green taxis, 20%, improperly identified as blue. Do the math and you get the 59% likelihood that the witness will be wrong. On the one hand, I find this counterintutive to the point of loveliness. It is true that the witness will always identify any specific taxi 4 times out of 5. It is true that this 4/5 success rate always stays the same no matter what the ratio of green to blue taxis. But at the same time it's equally true that altering the ratio of blue to green can make it probable that the witness will be wrong when she says "blue". That's very cool. But on the other hand, it now seems pretty obvious that 80% of 15 is smaller than 20% of 85. Go figure. Thanks everybody for a most enjoyable discussion.
• I'm glad you figured it out, after the requisite cooking time. You got what I was going to write, that, once we are given what the witness saw, we only use two of the four probabilities. We enumerate all four mainly as a standard way to organize all possibilities/probabilities before we actually apply the "it is given that the witness saw blue" part. That way we're ready for anything the problem throws at us. Thanks to you, too, and to all.
• On the one hand, I find this counterintutive to the point of loveliness. That's a brilliant summary of the thread, Torluath!
• Thanks for them bananes, Roryk. You're right that wolfram site, it rocks!!! From the article on the Weak Law of Large Numbers, I see that the base assumptions are: "Let X_1, ..., X_n be a sequence of independent and identically distributed random variables, each having a mean ==mu and standard deviation sigma." Whoa, this is so cool, it's scary. We're starting from randomness and ending up in a form of certainty. As stated in the book, this law gives you a number, N, of observations you must make to get a given degree of accuracy about the events in question. I read the Strong Law of Large Numbers, but I'm not sure what it actually means. What's the plain speech interpretation? Help!
• I find this counterintutive to the point of loveliness. That's a brilliant summary of the thread, Torluath! I agree, that's one of the more beautiful statements I've seen in a while.
• Ur, gosh. Thanks.
• This thread would make an excellent basis for a tutorial. Naturally I had a *cough* complete and confident grip of all the concepts personally, just saying useful stuff, so thanks.
• The strong law basically demonstrates that as your sample size increases, the probability that the sample mean and the population mean will be exactly equal approaches 1. This is fairly intuitive. Where all this becomes important is the Central Limit Theorem (Wolfram link), which allows us to make (probabilistic or confidence-based) statements about populations based on relatively small samples.
• On the weekend I witnessed a car accident. For real. A teenaged girl stepped out from behind a city bus, and basically just walked right in front of an oncoming car. Luckly it wasn't going very fast. The girl bounced off the front grill, landed on the pavement, got up on one elbow for a few seconds, then collapsed in a heap. Cars stopped everywhere -- we were on a busy, four lane road -- cell phones flashed out and the cops & paramedics were there in literally a minute or two. From what I could see, it looked like the girl was going to be ok. The car that hit her was green.
• Control your powers Torluath or we will have to organize a mob to burn you alive, and I don't want to do that.
• Sure you do.
• Don't tell him what he want, woman! An' where my samwich?!
• Listen I never intended to light Uri Geller on fire. How could I know freaks with psychoflex powers were so flammable.... when they are covered in kerosene.
• Thanks, Roryk!
• The car that hit her was green. I wasn't there but I can say with confidence that it was 80% green...carrying a big load of samwiches....QED.
• Come to think of it, my car's green, too. Move along, move along, nothing to see here...
• Good grief, i just read the same problem, the green taxis and blue taxis, in another book called "Inevitable Illusions", which talks about built-in cognitve errors and how our intuition can lead us badly astray. But for the life of me, i couldn't remember the way to get the exact answer! So i had to come back to this thread and read it over. argh.