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Unformatted text preview: MN1025 – Business Statistics 9 Lecture 3—Friday 25/1/2008 PROBABILITY (cont’d) PROBABILITY DISTRIBUTIONS Reference: Lind et al. , Chapters 5–7. 3.1 Lecture notes, Moodle, worksheets Please read the lecture notes. Please read the course information on the MN1025 course website, http://moodle.rhul.ac.uk. Please submit your solutions to the worksheets be- fore the deadline. 3.2 Conditional Probabilities We have already considered some simple probabili- ties. In practice we are often concerned with condi- tional probabilities , i.e. probabilities given that some- thing has happened, or given that someone is in a special category. Conditional probabilities arise when we restrict the set of events in some way. For example, consider the probability of a firm, taken at random, going bust in the next twelve months (this is event A and the probability is written P ( A )). Now what is the probability of a firm that has taken out a large bank loan (event B ) going bust? In this case we want a conditional probability P ( A | B ), the probability of A occurring given that B has occurred, that is the probability of a firm going bust given that it has taken out a large bank loan. Conditional probabili- ties can be quite subtle—it is easy to confuse P ( A | B ) with P ( B | A ), as you can see from the next examples. Read P ( A | B ) as “probability of A given B ”. 3.3 Example: Violent videos It is often suggested that watching violent videos leads to similar violent crime. As evidence we are told that a high proportion of those who commit violent crimes ( V C ) have recently watched violent videos ( V V ). So what we are given is P ( V V | V C ). But what is more important, but harder to calculate, is P ( V C | V V ): What is the probability that some- body who watches violent videos goes on to com- mit violent crime? These (conditional) probabilities could be very different. Here are some possible fig- ures for a large sample from the overall population. To compute the probabilities, we assume that a per- son is chosen at random from this sample. Did Did not Totals commit violent crime Watched violent videos 17 4583 4600 Did not watch 3 5397 5400 Totals 20 9980 10000 We find P ( V V | V C ) = 17 / 20 = 0 . 85, very high, but P ( V C | V V ) = 17 / 4600 = 0 . 0037, very small. A more useful quantity may be P ( V C | V V ) P ( V C | not V V ) = . 0037 . 0005556 = 6 . 66 , using P ( V C | not V V ) = 3 / 5400 = 0 . 0005556. So, based on this sample, those who watch violent videos are more than 6 times as likely to commit violent crimes than those who watch no such videos. 3.4 Example: Cancer screening Here are the results of a survey of 2000 women screened for some form of cancer. The first group of women had some symptoms prior to the screening, whereas the second group was free of symptoms. All the patients tested were checked later to see if they actually had cancer....
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This note was uploaded on 04/17/2008 for the course MN 1025 taught by Professor Schack during the Spring '08 term at Royal Holloway.
- Spring '08