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## Venn Diagrams and Set Operations: Tools for Probability

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**Venn Diagrams and Set Operations: Tools for**Probability**Properties of the Probability of an Event**• Here are some properties of the probability of an event that everyone must remember. • Let E be an event of a sample space S. • If E is the empty set, then P(E)=0. • For instance, if two dice are tossed, the probability that the sum of the faces that turn up is less than 2 is 0. • If E is the whole sample space S, then P(E)=1. • For instance, if two dice are tossed, the probability that the sum of the faces that turn up is between 2 and 12, inclusive, is 1. • Otherwise, 0<P(E)<1. • That is, the probability of an event is always positive and is never more than 1.**Sets and Venn diagrams can help us investigate other**interesting properties of the probability of an event.**Example #1**• 12,000 people voted for a politician in his first election, 15,000 voted for him in his second, and 3,000 voted for him in both elections. 55,000 people voted in the elections. • What is the probability that a randomly chosen voter voted for the politician • in at least 1 one of the elections? • in neither one of the elections?**We can extend the use of Venn diagrams by filling them with**probabilities of events in a sample space S. • But now, all the probabilities in the Venn diagram must add up to 1.**Example #2**• Let A and B be two events of a sample space S such that p(A)=0.45, p(B)=0.35, and p(A∩B)=0.15. • Use the given information to determine: • p(AUB) • p(A’∩B) • p(A’∩B’)**Venn diagrams can be quite helpful in solving real-life**probability problems, as the next example shows.**Example #3**• The manager of a repair shop has observed that a car will require a tune-up with a probability of 0.6, a brake job with a probability of 0.1, and both with a probability of 0.02. • What is the probability that a car will require • either a tune-up or a brake job? • a tune-up but not a brake job? • neither type of repair?**The Complementary Rule**If the probability of getting your dream job by age 30 is 0.25, then the probability of not getting it by that age is 0.75 • Thus if E is an event of a sample space S, then • p(E)=1-p(E’) • p(E’)=1-p(E) These results are referred to as the Complementary Rule**Example #4**• A bin in a bargain outlet contains 100 blank cassette tapes, of which 15 are known to be defective. • If a customer selects 20 of the tapes, determine the probability that at least 1 of them is defective.**The Complementary Rule that we have encountered is used in a**memorable mathematics problem: The Birthday Problem