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This 6 page Class Notes was uploaded by Clinton Beier on Wednesday September 23, 2015. The Class Notes belongs to HNRS200 at Drexel University taught by ScottKnowles in Fall. Since its upload, it has received 18 views. For similar materials see /class/212585/hnrs200-drexel-university in Honors Program at Drexel University.
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Date Created: 09/23/15
HNRS 200 Probability in the Universe Lecture 5 Notes Conditional Probability Thus far we have studied independant probability However in many situations past ob servations can be effectively used to predict the future or unknown situations In order to illustrate this idea let7s consider a simple example which has shown up in many books of brain teasers and also done in a fairly famous column by Marilyn Vos Savant who is alleged to have the highest 1Q in the world 77Suppose you7re on a game show and you7re given the choice of three doors Behind one door is a car behind the others goats You pick a door say No 1 and the host who knows what7s behind the other doors opens another door say No 3 which has a goat He then says to you 7Do you want to pick door No 277 ls it to your advantage to take the switch77 The answer surprisingly is yes Consider that when you rst select a door the odds are 13 of picking the car Let7s assume you always pick door 1 So the scenarios are 1 The car is behind door 1 You pick door 1 The host shows you door2 has a goat You switch to door 3 You lose 2 The car is behind door 2 You pick door 1 The host shows you door 3 has a goat You switch to door 2 You win 3 The car is behind door 3 You pick door 1 The host shows you door 2 has a goat You switch to door 3 You win In other words by taking the switching strategy you win 23 times as opposed to 13 by not switching Medical Results A more true to life example can be seen by interpreting the results of medical exames Let us imagine that you go into the doctors of ce and are given a routine test for a particular disease Now its important for this discussion that it be routine If you have any a priori ahead of time reason to suppose you have the disease then naturally that changes the nature of the ensuing discussion Further let7s imagine that only 1 of the population 101000 have this disease The medical company which makes the disease has issued the following table Cases per 1000 l l Has Disease l Doesn7t have disease l 1 Positive 1 9 l 50 l l Negative l 1 l 940 l How would the medical company describe the accuracy of their test Well they could say its 90 accurate After all 9 times out of 10 if you have the disease you will come up positive on the test The other 1 time when you have the disease and come up negative is known as a false negative They could also say that the test is 5 accurate After all of the roughly 990 people who don7t have the disease only 5 come up positive on the test These cases are known as false negatives Probability in the Universe Conditional Probabilityi 1 HNRS 200 Probability in the Universe Lecture 5 Notes But in reality the odds are somewhat different What you7d really like to know is not What are the odds given that you have the disease of having a positive result on the test But rather What are the odds given that you have a positive result on the test of actually having the disease I want you to think about the difference between the two statements for a bit because they really are all the difference in the world Consider that out of every 1000 people who take the test 59 of them get a positive However of those only 9 really have the disease 95915 In other words even though you tested positive the odds are actually quite low that you have the disease Bayes7 Theorem Statisticians have a fancy rule that they use to describe posterior probabilities It is known as Bayes7 Theorem As an equation it reads PA B P A B 77 1 lt l gt PUB lt gt The term PAlB means the probability that some statement A is true in our example 7 You have the disease given that you already know that statement B is true in our example 7 You tested positve The term PA B is the probability that both are true And PB is the a priori probability that B would come up true Whats the probability of having the disease and coming up positive 09 Whats the a priori probability of coming up positive 59 And thus the probability of having the disease given that you came up positive is 095915 Exactly as we found before DNA Evidence The reason I bring all of this up is that conditional probability is one of the most misun derstood and misused forms of statistics One particularly pervasive example is in the use of DNA evidence Consider the following A defendant John Doe has been arrested for murder A bit of blood was found on the murder weapon and after doing a DNA test the results came back a perfect match Moreover the lab report said that only 11 Million people in the country would have come up as a match on this test So the prosecuter makes his case Using only the DNA evidence the probability is 9999991000000 that Mr Doe is guilty Since the standard is beyond a reasonable doubt77 the jury has no choice but to convict The defense makes the argument No There are 300000000 people in the US That means that 300 would have the same DNA and thus the probability given that only 1 of them actually did commit the crime is only 1300 about 03 and thus the defendant should be aquitted Who7s right Probability in the Universe Conditional Probabilityi 2 HNRS 2007 Probability in the Universe Lecture 5 Notes ln reality7 we realize that there is an unspoken assurnption7 one that puts mathematics and legality at odds O icially7 a defendant is presumed innocent until proven guilty ln reality7 though7 we assume that because someone has been arrested that there is some reasonable chance that they are guilty So the question is What is the prior probability that a person on the stand is innocent If we call it Pl7 then7 and P is the probability of testing positive on the DNA test PUH 0000001 gtlt PU 0000001 gtlt PI17 PI where 1 Pl is simply the a priori probaility that the defendant based on other evidence is guilty He must be one or the other Consider the cases where based on other evidence7 the chance that hes innocent is 0999 In that case7 the DNA evidence overwhelrningly changes sour view7 and the posterior probability of his innocence is merely 11000 Easily enough to convict O N O 4gt 0 m 0 in H4 Probability in the Universe Conditional Probabilityi 3 HNRS 200 Probability in the Universe Lecture 3 Notes Why do Casinos make money Casinos are a tremendous money making venture Everybody knows this And despite other delusions that people may have about a particular system of betting it is nevertheless true that with the exception of games like poker where you play against other civilians you are more likely to lose than to win So why are people still willing to play The easy answers are that people are dumb or uninformed or that they are simply paying for entertainmen and certainly all of these are true to some degree but consider a game where you lost every time Would you still play Let7s consider a fairly simple game 7 Roulette Now in Roulette there is a wheel which has slots labeled from 1 36 half in red and half in black as well as a slot with a green 0 and another green 00 A ball is released and falls randomly in one of the slots This game unlike many others card games especially has no memory The outcome of a particular spin is completely independent of what happened previously One of the most common bets consists of selecting red or black Now it is clear to see that since there are a total of 38 slots and 18 of them are say red then the probability of red coming up is i 18 i 0 473 p 7 38 7 39 Expectation Value This means that if you bet say a dollar on a given spin then the expectation value simply put the return for each outcome multiplied times the probability of the outcome and all added up of your return is Er p gtlt117pgtlt71 70054 This means of course that on average for every dollar 1 put down I will lose about a nickel each bet If that loss were steady then ultimately all of your money would be depleted But wait This is a random process and therefore it behaves like a random walk Consider that if you bet 1 dollar N times in a row then your expected loss will be 7005 X NixN In other words if you play 10 spins and bet on red each time then sure on average you7ll lose 50 cents 005 X 10 but your range reasonable range of outcomes will be between 366 to 266 This means that fairly often you7ll come out a winner Let7s consider the situation generally for a game which pays even odds ln roulette your expected return is 5 in Blackjack with typical rules and assuming you play optimally your expected return is about 05 per hand In both cases though the standard deviation in your return will be 7W 0 is the standard deviation per hand in this particular case 0 1 That means that if you play for a long time a reasonable N 70 of the time range of outcomes will be Tet MN i am Now lets say you7re lucky You7re at the upper limit of your expected performance You7re ahead if your return is above zero The longer you play the less likely that7s to be the case Probability in the Universe Gamblingi 1 HNRS 200 Probability in the Universe Lecture 3 Notes Consider the upper limit ret MN IxN if you7ve played for a very long time the best you can reasonably hope for is to exactly break even How long MN O39VN 0 a 2 M In other words in Roulette you can play for about 400 spins before you can be pretty certain of losing ln blackjack its more like 40000 hands And this in essence is why casinos make money Consider that in a typical visit l7m only likely to play say 100 hands of blackjack On average l7m only going to lose 12 of a hand more than I win And the scatter is something like 10 hands in either direction In essence this means that something like 45 of the time l7m going home a winner And that7s great for the casino because it means that l7m more likely to come back next time Meanwhile since the casino is averaging over so many players they consistently make returns equal to the house edge Long Odds There7s another type of bet that7s available in Vegas or in your lottery for example and that7s one with long odds Besides being able to bet on red vs black Roulette also o ers you the opportunity to bet that a speci c number will come up If that happens you get 35 times your bet Clearly the expected return is or Em 35 X 138 71 X 3738 70053 A very similar return but the vast majority of the time you lose and only occasionally do you win This means that for reasons l7m not going to go into the standard deviation in your return is something like 58 X W and thus there is a much larger scatter over many many spins of 7005 X N 58 gtlt xN So even though in the long run you do about as well betting a single number or betting red vs black and in both cases you lose the noise is much larger Consider that while 400 spins of red vs black will guarantee you to lose it7ll take something like 13000 spins of picking a particular number before you7re guaranteed to lose Of course on the opposite side when you lose you7re likely to lose much more under this scenario The longer the odds even if the return is the same the longer you can play without being guaranteed to lose As a particularly salient example consider slot machines In many casinos slot machines are set to return 12 on average lf slots returned a simple even odds bet similar to red vs black then after 100 pulls of the slot machine which you could probably accomplish in just under 5 minutes the odds of being even or better would be something like 18 After Probability in the Universe Gamblingi 2 HNRS 200 Probability in the Universe Lecture 3 Notes twice as long the odds would by about 34 How on earth could people keep playing for hours if they were guaranteed to lose after such a short time7 The answer is that a typical return from a slot machine is one that pays say 1001 However that happens very very infrequently say 1 in 110 pulls As a result while a typical player will still lose the lucky ones can end up ahead The losers however can lose everything Risk of Ruin Which brings us to the concept of risk of ruin77 Remember gambling is a random walk Imagine that you are betting 10bet and you have 100 in your pocket It doesn t matter if you eventually would have beaten the house if at any point you drop below 710 bets Once you re out of money you re out of money The casino of course has a much looser de nition of what being out of money77 means and they thus are kind enough to provide many opportunities for getting more ATMs credit card cash advances and the like at various convenient places The basic argument still holds though At some point you will no longer be able to borrow any more cash and you re done 1 l l l l m 0 m wquotquot m 0 son mm m an Even in a fair eveniodds game you have a very high probability of simply running out of money If you start with say 5 bet units eventually there s something like an 85 probability that you will eventually have a downturn and run out of cash Probability in the Universe Gamblinge 3
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