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

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4. Probability Rules and Conditional Probability 4.1 General Methods In the mathematical definition 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 definitions 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 specific 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 finishing 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 finishing 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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30 AB S A B A B A B A B S S A complement A complement A A B Figure 4.3: Top panel: A B means A OR B (or possibly both) occurs. A B is shaded. Middle panel: A B (usually written as AB in probability) means A and B both occur. A B is shaded Lower panel: ¯ A means A does not occur. ¯ A is shaded Let A = { math average 80% } B = { overall average 80% } C = { STAT 230 80% } In terms of these symbols, we are given P ( A )= . 22 ,P ( B . 20 ( C . 24 ( AC . 14 ( BC . 13 ( ABC . 1 ,and P ( ¯ A ¯ B ¯ C . 67 .Weareaskedto find P ( AB ¯ C ) , the shaded region in Figure 4.4 Filling in this information on a Venn diagram, in the order indicated by (1), (2), (3), etc. (1) given (2) P ( AC ) P ( ABC ) (3) P ( ) P ( ABC ) (4) P ( C ) P ( AC ) . 03 (5) unknown (6) P ( A ) P ( AC ) x (7) P ( B ) P ( ) x (8) given
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31 Figure 4.4: Venn Diagram for Math Averages Example (Usually, we start filling in at the centre and work our way out.) Adding all probabilities and noting that P ( S )=1 , we can solve to get x = . 06 = P ( AB ¯ C ) . Problems: 4.1.1 In a typical year, 20% of the days have a high temperature > 22 o C. On 40% of these days there is no rain. In the rest of the year, when the high temperature 22 o C, 70% of the days have no rain. What percent of days in the year have rain and a high temperature 22 o C?
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chap4 - 4. Probability Rules and Conditional Probability...

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