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Chapter 6. Section 6.3 Part 1 – Probability Rules. Probability Rules. Recall the following probability rules from sec. 6.2: 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 Section 6.3 Part 1 – Probability Rules
Probability Rules • Recall the following probability rules from sec. 6.2: • 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 • 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, • 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 B are independent, then
Addition Rule For Disjoint Events • The general probability rules can be extended to more than two events. • Addition rule for disjoint events: • If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then • This rule extends to any number of disjoint events.
General Addition Rule for Unions of Two Events • The union(represented by the word “or”) of any collection of events is the event that at least one of the collection occurs. • If events are not disjoint, they can occur simultaneously. • The following diagram suggests that the outcomes common to both are counted twice when adding the probabilities, so we must subtract the probability once.
General Addition Rule for Unions of Two Events • The following is the addition rule for the union of any two events, disjoint or not: • For any two events A and B, or • Note – even though the event {A and B} cannot happen for disjoint events, it is still included in the formula since it’s an empty event that wouldn’t affect the probabilities anyway.
Example • To improve tourism between France and the U.S., the two governments form a committee consisting of 20 people: 2 American men, 4 French men, 6 American Women, and 8 French women. If you meet one of these people at random, what is the probability that the person will be either a woman or a French person? • Also see example 6.17 on p.362-363 probability of a French woman probability of a woman probability of a French person
Joint Probability • A jointevent is a simultaneous occurrence of two events. • The probability of a joint event is called a joint probability. • Another way to work with joint events (other than a Venn diagram) is to use tables.
Conditional Probability • Conditional probability – gives the probability of one event under the condition that we know another event. • Represented by the notation ) and read as “the probability of A given the information B” • Ex. – the probability that the next card dealt is an ace given that exactly one of the four visible cards is an ace.
General Multiplication Rule forany Two Events • The probability that both of two events A and B happen together can be found by: • Here is the conditional probability that B occurs given the information that A occurs. • In words, this rule says that for both of two events to occur, first one must occur and then, given that the first event has occurred, the second must occur. • See example 6.20 on p.368
Definition of conditional probability • If you know P(A) and P(A and B) you can rearrange the multiplication rule to produce a definition of the conditional probability in terms of unconditional probabilities: • When P(A) > 0, the conditional probability of B given Ais: • See example 6.21 on p.369
