chap4 - 4 Probability Rules and Conditional Probability 4.1...

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4. Probability Rules and Conditional Probability 4.1 General Methods In the mathematical de fi nition of probability, an arbitrary event A is merely some subset of the sample space S . The following rules hold: 1. P ( S ) = 1 2. For any event A, 0 P ( A ) 1 It is also obvious from our de fi nitions in Chapter 2 that if A and B are two events with A B (that is, all of the simple events in A are also in B ), then P ( A ) P ( B ) . It is often helpful to use elementary ideas of set theory in dealing with probability; as we show in this chapter, this allows certain rules or propositions about probability to be proved. Before going on to speci fi c rules, we’ll review Venn diagrams for sets. In the drawings below, think of all points in S being contained in the rectangle, and those points where particular events occur being contained in circles. We begin by considering the union ( A B ) , intersection ( A B ) and complement ( ¯ A ) of sets (see Figure 4.3). At the URL http://stat-www.berkeley.edu/users/stark/Java/Venn.htm, there is an interesting applet which allows you to vary the area of the intersection and construct Venn diagrams for a variety of purposes. Example: Suppose for students fi nishing 2A Math that 22% have a math average 80%, 24% have a STAT 230 mark 80%, 20% have an overall average 80%, 14% have both a math average and STAT 230 80%, 13% have both an overall average and STAT 230 80%, 10% have all 3 of these averages 80%, and 67% have none of these 3 averages 80%. Find the probability a randomly chosen math student fi nishing 2A has math and overall averages both 80% and STAT 230 < 80%. Solution: When using rules of probability it is generally helpful to begin by labeling the events of interest. 29
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