P9 p a b p a p b p a b a venn diagram makes this

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(p9) P A B P A P B P A B . A Venn diagram makes this clear. If A and B intersect, adding P A and P B double counts the outcomes in both A and B . A B A B 36
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Because a probability is no greater than one, we (p3) implies that i 1 P A i 1 when A 1 , A 2 ,... is a mutually exclusive set of events. Recall that the infinite sum is actually the limit of the sums i 1 n P A i as n . Because each P A i 0, infinite sums involving probabilities are well defined. For non-mutually exclusive events, we could have i 1 P A i 1, or even i 1 P A i [which means the sequence of partial sums i 1 n P A i is unbounded]. 37
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(p10) For any sequence of events A j : j 1,2,... , P j 1 A j j 1 P A j . This upper bound for P j 1 A j is not always useful. For example, if A j   for all j then we conclude P ≤  , which is no information at all. If A 1 and A 2 are two events with probability 1/2, the inequality is P A 1 A 2 1, which is of no value. 38
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Given a measurable space , (sample space, -field) and a probability measure defined on this space, we call , , P a probability space . For computing probabilities in applications, we often rely on intuition or, more formally, a “frequentist” approach to assigning probabilities to basic events. This can be thought experiment about what would happen if the experiment were done over and over again or it can be based on actual experience. 39
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When we turn to statistics, we will use data to estimate probability models , and then we can estimate probabilities of different events. Given probabilities of simple events we can apply the various rules of probability to calculate the probability of more complicated events. 40
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