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Unformatted text preview: Probability I Probability
Read Chapter 16 Recall the discrimination experiment from last time… experiment
Percent Percent Positive Positive Label No Label 24% 70% Total N 90 90 180 Thought questions: What is the probability that a coin flipped What lands heads? lands What is the probability that Michigan State What will win the NCAA men’s basketball championships? championships? Interpretations of Probability Interpretations Long-run relative frequency: the Long-run proportion of time it occurs over the long run run Personal probability: the degree to which a Personal given individual believes an event will happen happen The previous questions show that each is The appropriate to a different usage of the term “probability” term Relative frequency interpretation Relative How to calculate this type of probability? How For example, what is the probability that a For baby born will be a male? baby A) make an assumption
• E.g. ½ probability of a male, ½ female Two ways to calculate the probability: B) observe the long term frequency
• .512 of births in the US are male Relative-Frequency Interpretation Relative-Frequency Can be applied in a situations that will be Can repeated a number of times repeated The frequency settles on a single The constant value over the long run; this is a the probability the Cannot be used to calculate the Cannot probability of a one shot event (e.g. probability MSU makes it to the Final Four in 2010) in Personal probability Personal Applies to situations that will never occur again Cannot use frequency data to calculate these Based on personal experience What is the probability that: The New England Patriots win the Super Bowl in 2011? the world will end by nuclear war in the next 50 years? the Cannot use frequency data to calculate these; must use Cannot knowledge of politics, economics, sports, etc. knowledge May differ from person to person, but must be coherent Rules Rules Regardless of the interpretation, Regardless probabilities must obey certain rules probabilities Rule 1 Rule If there are two possible outcomes, the If probabilities must sum to 1 probabilities Probability of head, tails in coin flip Probability of male, female birth Probability of an individual receiving a Probability negative, positive response in asking about a rental property rental Probability the world ends in the next 50 Probability years, the probability it does not years, Rule 1 Rule If you are on a jury, and your personal If probability that the defendant is guilty is .8, then the probability that she is not guilty must be .2 must Rule 2 Rule If two outcomes cannot happen If simultaneously, they are called mutually exclusive exclusive The probability of either of these outcomes The either occurring is the sum of their probabilities sum Probability of H, T in coin flip =.5 + .5=1 Probability of male or female: .51 + .49=1 The probability that you take a job in the The US after graduation is .4 US The probability that you travel abroad is .1 Then the probability of either of these Then must be .5 must We can add because they are mutually We exclusive exclusive But you can’t add probabilities if they are But not mutually exclusive: not Probability that the US bombs Iran by the end Probability of the year of Probability that Israel bombs Iran by the end Probability of the year of Rule 3 Rule If two events do not influence each other, and if If knowledge of one does not influence knowledge of the other, the events are independent independent Two separate coin flips 1st birth gender, 2nd birth gender But consider this case. You are dealt two cards. The But probability that the first is a club and the probability that the second is a club are NOT independent. Rule 3 Rule If two events are independent, the probability that they If both happen is found by multiplying them both 1st child male, second female=.51*.49=.2499 1st coin flip heads, 2nd heads=.5 * .5=.25 Why is this? Think of multiplying fractions. Half of one Why half is .5 * .5=.25. The same thing is true of probabilities: if there is a .5 chance of an event in the first round, a .5 in the second round, and the two are independent, then you multiply the two. Example Example Let’s assume that 20% of students smoke Let’s here (this is probably not too far off; 28% of high school seniors smoke) of Let’s assume everyone is randomly Let’s assigned to dorm rooms assigned What is the probability that a given dorm What room contains two smokers? We are calculating the probability that the first We roommate is a smoker AND the second roommate is a smoker roommate These two events (the first roommate is a These smoker, the second roommate is a smoker) are independent independent So we can multiply probabilities .20 * .20= .04 4% probability that the room will contain two smokers This rule does NOT work if the events are This not independent not E.g. Probability 1st card club, 2nd card club A more meaningful example of assuming independence when it does not hold… does Sally Clark was accused of murdering her 11-week old son Sally Christopher in 1996 and her eight-week old son Harry in 1998. Clark’s defense was that the children died from SIDS. But Dr. Roy Meadow testified at her trial that the odds of the two But boys dying from SIDS was “one in 73 million.” One case of SIDS is about 12/100,000 Meadow said the probability of 2 SIDS deaths is 12/100,000 * Meadow 12/100,000 12/100,000 This is astronomically low, so the expert witness concluded that This Clark must have murdered the boys Clark “This approach is, in general, statistically invalid. This It would only be valid if SIDS cases arose independently within families, an assumption that would need to be justified empirically. Not only was no such empirical justification provided in the case, but there are very strong a priori reasons for supposing that the assumption will be false. There may well be unknown genetic or environmental factors that predispose families to SIDS, so that a second case within the family becomes much more likely.” Royal Statistical Society Society Rule 4 Rule If the way in which one event can occur is a subset of If those in which another can occur, then the probability of the subset event cannot be higher than the probability of the one for which it is a subset the Probability of H on first flip Probability of H on first flip and H on second flip The probability of random draw from population a The Republican Republican The probability of random draw from population has The income greater than $70,000 AND is Republican income Assign a probability to each of these Assign occurring in the next 50 years: occurring The world will come to an end The _______________ _______________ The earth will be destroyed by a meteorite or a nuclear war:_________ or Putting it all together Putting 1. if two possible outcomes, probabilities must 1. sum to 1 sum 2. probability of either of two mutually exclusive 2. outcomes is the sum of the probabilities outcomes 3. probability of both of two independent events 3. is the product of the two probabilities is 4. probabilities of subset events are no greater 4. than the probability of the event for which it is a subset subset Putting it all together Putting The probability of getting infected with HIV The from a single heterosexual encounter without a condom with unknown partner is at most 1/500 at What is the probability of getting infected What after 4 such encounters? First, we will calculate the probability of First, getting infected during each of the four encounters if one has not received the infection before infection Second, we can add these together to get Second, the cumulative probability—that is the probability of getting infected up to any given time given Probability of FIRST Infection Probability 1st encounter = .002 2nd encounter (and not on earlier encounter)=.998*.002=.001996 encounter)=.998*.002=.001996 3rd encounter (and not on previous two encounters)=.998*.998*.002=.001992 encounters)=.998*.998*.002=.001992 4th encounter (and not on previous three encounters)= encounters)= .998*.998*.998*.002=.001988 .002 + .001996 + .001992 + .001988 = . 007976 Probability is FUN!!! Probability intrade.com A proposal… proposal… Someone offers you a bet on the MSU – Penn State Someone football game this year. football If MSU wins, you win $100. If they lose, you lose $100. If Do you accept? Some things to consider: What if you stand to lose $1000 (but still only stand to What gain $100)? What if you stand to lose $10,000? gain Expected Value Expected The expected value is the average benefit The of any measurement over the long run of Amounts associated with each outcome: Amounts A1, A2, A3 …Ak for k outcomes A1, Probabilities of receiving each amount=p1, Probabilities p2, p3…pk p2, Expected Value (EV)=A1*p1 + A2*p2 + Expected A3*p3 +…Ak*pk for k amounts. A3*p3 What is the expected value of the California What Decco lottery game? Decco First, we need to know the net value of each First, net outcome; this is the relevant Amount for each outcome outcome $5000 $50 $50 $5 $5 $0 1/28,561 1/595 1/595 .0303 .0303 .726 We have to subtract $1 for the cost of the ticket from We each prize for the net amount you would get for each outcome outcome $4999 $49 $49 $4 $4 $-1 1/28,561 1/595 1/595 .0303 .0303 .726 EV= $4999*1/28,561 + $49 * 1/595 + $4*.0303 + $1*.726 = -$.35. Compare this with an EV of 0 if we do not play Expected value theory is both a normative Expected and a descriptive theory and This is useful in many contexts For example, what is the long run value of a For given investment? Should you bet on a sports team? team? If we allow that the benefits of different If nonmonetary outcomes, then this theory is widely applicable. widely
• Should I propose to my girlfriend? Should we have kids? Should Should I major in journalism or engineering? Etc. http://www.lasvegassun.com/gambling-addictio http://www.lasvegassun.com/gambling-addicti Sounds useful, right? People are horrible at thinking about this People stuff…. stuff…. More on this next time. Questions? ...
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This note was uploaded on 01/27/2011 for the course COM 200 taught by Professor Tamborini during the Fall '09 term at Michigan State University.
- Fall '09