PROBABILITY THEORY:CONDITIONAL PROBABILITY AND INDEPENDENT EVENTS

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CONDITIONAL PROBABILITY AND INDEPENDENT EVENTS

Often experiments are performed so that some results are obtained early in time and some later in time. This is the case, for example, when the experiment consists of measuring the demand for a product during each of 2 months; the demand during the first month is observed at the end of the first month. Similarly, the arrival times of the first two customers on each of 2 days are observed sequentially in time. This early information can be useful in making predictions about the subsequent results of the experiment. Such in- formation need not necessarily be associated with time. If the demand for two products during a month is investigated, knowing the demand of one may be useful in assessing the demand for the other. To utilize this information the concept of “conditional probability,” defined over events occurring in the sample space, is introduced.

Consider two events in the sample space E1 and E2, where E1 represents the event that has occurred, and E2 represents the event whose occurrence or nonoccurrence is of interest. Furthermore, assume that P{E1} > 0. The conditional probability of the occur- rence of the event E2, given that the event E1 has occurred, P{E2½E1}, is defined to be

INTRODUCTION TO OPERATIONS RESEARCH-0549

where {E1 n E2} represents the event consisting of all points w in the sample space common to both E1 and E2. For example, consider the experiment that consists of observing the demand for a product over each of 2 months. Suppose the sample space consists of all points (x1,x2), where x1 represents the demand during the first month, and x2 represents the demand during the second month, x1, x2 0, 1, 2, . . . , 99. Furthermore, it is known that the demand during the first month has been 10. Hence, the event E1, which consists of the points (10,0), (10,1), (10,2), . . . , (10,99), has occurred. Consider the event E2, which represents a demand for the product in the second month that does not exceed 1 unit. This event consists of the points (0,0), (1,0), (2,0), . . . , (10,0), . . . , (99,0), (0,1), (1,1), (2,1), . . . , (10,1), . . . , (99,1). The event {E1 E2} consists of the points (10,0) and (10,1). Hence, the probability of a demand which does not exceed 1 unit in the sec- ond month, given that a demand of 10 units occurred during the first month, that is, P{E2⏐E1}, is given by

INTRODUCTION TO OPERATIONS RESEARCH-0550

The concept of conditional probability was introduced so that advantage could be taken of information about the occurrence or nonoccurrence of events. It is conceivable that information about the occurrence of the event E1 yields no information about the oc- currence or nonoccurrence of the event E2. If P{E2⏐E1} P{E2}, or P{E1⏐E2} P{E1}, then E1 and E2 are said to be independent events. It then follows that if E1 and E2 are independent and P{E1} 0, then P{E2⏐E1} P{E1 E2}/P{E1} P{E2}, so that P{E1 E2} P{E1} P{E2}. This can be taken as an alternative definition of independence of the events E1 and E2. It is usually difficult to show that events are independent by using the definition of independence. Instead, it is generally simpler to use the information avail- able about the experiment to postulate whether events are independent. This is usually based upon physical considerations. For example, if the demand for a product during a month is “known” not to affect the demand in subsequent months, then the events E1 and E2 defined previously can be said to be independent, in which case

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Intuitively, this implies that knowledge of the occurrence of any of these events has no effect on the probability of occurrence of any other event.

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