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Chapter 5 Conditional probability In this chapter, we look at more complicated notions of probability, and extend the multiplica- tion rule for probability to cater for events that are not independent. 5.1 Introduction So far we have only considered probabilities of single events or of several independent events, like two rolls of a die. However, in reality, many events are related. For example, the probability of it raining in 5 minutes time is dependent on whether or not it is raining now. We need a mathematical notation to capture how the probability of one event depends on other events taking place. We do this as follows. Consider two events A and B . We write P ( A | B ) for the probability of A given that B has already happened. We describe P ( A | B ) as the con- ditional probability of A given B . For example, the probability of it raining in 5 minutes time given that it is raining now would be P ( Rain in 5 minutes | Raining now ) . Example Utility companies need to be able to forecast periods of high demand. They describe their fore- casts in terms of probabilities. Gas and electricity suppliers might relate them to air temperature. For example, P ( High demand | air temperature is below normal ) = 0 . 6 P ( High demand | air temperature is normal ) = 0 . 2 P ( High demand | air temperature is above normal ) = 0 . 05 . 54
CHAPTER 5. CONDITIONAL PROBABILITY 55 We can calculate these conditional probabilities using the formula P ( A | B ) = P ( A and B ) P ( B ) , that is, in terms of the probability of both events occurring, P ( A and B ) , and the probability of the event that has already taken place, P ( B ) . To see how this formula works, let’s consider the class of students from Exercises 4 (on page 53). Student Height Weight Shoe Student Height Weight Shoe Number Sex (m) (kg) Size Number Sex (m) (kg) Size 1 M 1.91 70 11.0 10 M 1.78 76 8.5 2 F 1.73 89 6.5 11 M 1.88 64 9.0 3 M 1.73 73 7.0 12 M 1.88 83 9.0 4 M 1.63 54 8.0 13 M 1.70 55 8.0 5 F 1.73 58 6.5 14 M 1.76 57 8.0 6 M 1.70 60 8.0 15 M 1.78 60 8.0 7 M 1.82 76 10.0 16 F 1.52 45 3.5 8 M 1.67 54 7.5 17 M 1.80 67 7.5 9 F 1.55 47 4.0 18 M 1.92 83 12.0 Suppose we want the probability that a student chosen at random from this class will be female given that the student’s shoe size is less than 8. We could simply find the proportion of students with shoe sizes less than 8 who are female. There are 7 students with shoe sizes less than 8 and 4 of these are female. So P ( Female | Shoe size less than 8 ) = 4 7 . This probability can also be calculated using the above formula as follows: P ( Shoe size less than 8 ) = 7 18 , P ( Shoe size less than 8 and female ) = 4 18 ; and so P ( Female | Shoe size less than 8 ) = P ( Shoe size less than 8 and female ) P ( Shoe size less than 8 ) = 4 / 18 7 / 18 = 4 7

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