# Substituting b for p a b gives 0 p b p a p b p a b p

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Substituting B for ( ) P A B gives ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) P B P A P B P A B P A P A B P A P A B An even simpler proof: Since A B , the event ( ) ( ) A B A P A B P A . So, when it comes to the intersection of two events, 0 ( ) min{ ( ), ( )} P A B P A P B Theorem 2.6: ( ) ( ) ( ) ( ) P A B P A P B P A B Proof: This is an algebraic manipulation of our union theorem. It seems clear that we will want to revisit the probability of the intersection of two events later.
8 2.5 Probability Viewed as a Relative Frequency While the definition of a probability function has been defined mathematically as a function that follows a set of rules, we seek to put some practical meaning to probability. This might make studying probability more interesting. It would allow us to use this field of study to get a better understanding of real-world situations. Suppose that we have a finite sample space with n-outcomes. We could make the following assignment of probability for each outcome in : 1 i P O n for 1,2,..., i n . This is clearly a probability function on our sample space and this function assigns an equal probability to each outcome in . Exercise 2.6: Confirm the probability as defined above conforms to the requirements in our definition of a probability function. Example 2.15: Suppose that we roll a single die. Since {1,2,3,4,5,6} we have ( ) 1 / 6 i P O . If we now think of probability as the “chance” that a specific outcome is selected, w e might say that each outcome is “equally likely” to be selected since each i O has the same chance of occurring. That is, each outcome is equally as likely as any other outcome to occur. Based on our lifetime of experience of referencing the words chance, probability and likelihood, we should start feeling more comfortable about this recently defined function called a probability function. Note that a sample space does not need to have a finite number of outcomes. But it does have to if we want to apply the equally likely outcome concept to our probability assessment (assignment). A finite number of outcomes and an equally likely outcome assumption are common enough that this probability function is important enough to merit its own definition: Definition 2.13: The Equally Likely Probability Function for a finite sample space consisting of n- elements, each denoted by i O , is defined by 1 i P O n for {1,2,..., } i n Note that just because a sample space has a finite number of outcomes and it is legal to assign ( ) 1 / i P O n , it may be an unwise assignment if equally likely goes against our thought process of chance. For example, consider purchasing a lottery ticket and have our sample space consist of two basic outcomes { I win, I don't win} . Using the equally likely probability function, we would assign ( ) 1 / 2 P I win and ( ) 1 / 2 P I don't win . While legal under the rules of probability functions, if you’ve ever purchased lottery tickets before, you’ll know that this probability function is unreasonable in nearly every lottery.
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