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Chapter 6

Chapter 6. Section 6.2 Part 2 – Probability Rules. Probability Rules. Rule 1 : The probability of any event A satisfies: Rule 2 : If S is the sample space in probability model, then: Rule 3 :

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Chapter 6

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  1. Chapter 6 Section 6.2 Part 2 – Probability Rules

  2. Probability Rules • Rule 1 : • The probability of any event A satisfies: • Rule 2 : • If S is the sample space in probability model, then: • Rule 3 : • The compliment of any event A is the event that A does not occur, written as . The compliment rule states that: • The compliment of an event can also be represented as: or

  3. Probability Rules • Rule 4 : • Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in common and so can never occur simultaneously. • If A and B are disjoint then, • This is the addition rule for disjoint events. • Rule 5 : • Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. • If A and Bare independent, then • This is the multiplication rule for independent events.

  4. Set Notation • – read “A union B” is the set of all outcomes that are either in A or B. • – read “Aintersect B” is the set of all outcomes that are in Aand B. • Empty event – The event that has no outcomes in it. • If two events A and B are disjoint (mutually exclusively), we can write , read “A intersect Bis empty.”

  5. Venn Diagram For Disjoint Events • The following picture shows the sample space S as a rectangular area and events as areas within S is called a Venn diagram. • The events A and B are disjoint because they do not overlap; that is, they have no outcomes in common.

  6. Compliment • The compliment in the diagram below contains exactly the outcomes not in A. • Note that we could write • See examples 6.8 and 6.9 on p.344-345

  7. Venn diagram of independent events • Suppose that you toss a coin twice. You are counting heads so two events of interest are: • The events A and B are not disjoint. They occur together whenever both tosses give heads. • The Venn diagram illustrates the event {A and B} as the overlapping area that is common to both Aand B. • See example 6.12 on p.351 and example 6.14 on p.353

  8. Independent and Disjoint • Be careful not to confuse disjointness with independence • Recall that disjoint events (or mutually exclusive events) tell us that if event A occurs that event B cannot occur • With independent events the outcome of one trial must not influence the outcome of any other • For example: • A subject in a study cannot be both male and female, nor can they be aged 20 and 30. A subject could however be both male and 20, or both female and 30. • Unlike disjointness or compliments, independence cannot be pictured by a Venn diagram, because it involves the probabilities of the events rather than just the outcomes that make up the events.

  9. Homework: p.348-358 #’s 19, 20, 23, 26 (ignore Benfords Law, just use the probabilities), 28, 30, 31, & 41

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