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QPT Merged Files - Section 4.7 Complementagg Events Def 4...

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Unformatted text preview: Section 4.7 Complementagg Events Def 4: The complement of event A, denoted by A and read as “A Bar” or “A complement”, is the event that includes all the outcomes for an experiment that are not in A. EventA and A are complements of each other. The following Venn diagram shows the complementary events A and A . S “ml Properties: P(A)+ PG) = 1 . From this equation, we can deduce that P(A)=1_P(Z) and P(A)=l—P(A). Section 4.8 Intersection of Events and the Multi lication Rule: Section 4.9 Union of Events and the Addition Rule Def 5: The intersection of events A and 3 represents the collection of all outcomes that are common to both A and B and is denoted by A and B or by A n B. Def 6: The union of events A and B is the collection of all outcomes that belong either toA or to B or to both A and B and is denoted byA or B or byAUB. Venn Diagram: S - - AorB (AUB) AandB (AHBJ Calculating the probabilities of A U B and A H B The Multiplication Rule Used to calculate the joint probability P( A F] B): P(A)- P(B] A) . Conditional probability: if A and B are two events, then P(A|’]B) and P(A|B)=P(AHB) P(B | A): P (A) PU?) given that 19(44):t 0 and P(B) qt 0. If events A and B are independent, then the rule can be simplified to HAD 3) = P(A)-P(B). If more events are independent, then the simplified rule also holds. 1.13M Used to calculate the probability of the union of two events P(A.U B) = P(A)+ P(B)— PM 0 B). If there are more than two mutually exclusive events, then the simplified rule also holds. Ex 5: Consider the die-toss experiment. Define the following events: A:{ Toss an even number} B: {Toss a number less than 3} 3. Describe AU B b. Describe A f] B c. Calculate P( A U B) and P( A n B) assuming the die is fair. Using the formulas in sections 4.7-4.9, we can solve the following problems. Ex 6: A woman ’s clothing store owner buys from three companies: A, B and C. The most recent purchases are shown here. ___-_— m— — T—shirts ___- — Pants — _. 2‘0“ _ ___ _‘ __ p=€AfifliLté§rf _ __ _. _______ _ __ __‘V‘m . A=_?Hgflf3__. __ ___. _ - _ _ j=fr¢frfi __ “”4f [flAErF‘E‘AZQA of gram/1 (C 4&Mm/I/‘fgv4414-Q*—_ fro baLv'ch“, _ c 1. _ _, an!” :> .Mr‘owu: (Lien '_' a‘ H .... ‘ ’ I} _‘0{ >__ Mlfl_é______..__ _ fi. _ ..._.___ 30339}; HT, _fo !/_;eM RE E? M WMM _' _ PYAWE)__:M) ftp/59"Wm5) _ _a_A= e34} _ @ 5528:2422??? _' 5 J? H. _ ____ f: Lax/3&3 _ b) Mwief __ 2) mmgmns) f— — -- — —— -- —- fi—flf—amiéeawr'4—-Mg——_ it - —-- —— — ‘— rflQ UK): id: 01“- 0LA+COM65 :3 5— - h— 9— y if — — .3 —H-'_hl€'_ "fitté-fiflfln-e r {A ”Pang 2% . a f?) m 2%; 0:4 . g: . 'FZZ)’ET:WI‘)'=7*‘01 «m p r a 9rd,; T Maifafan _ '_"' __ g“ $461 “' or %:M Lectures Five and Six Section 4.4 Marginal and Conditional Probabilities Ex 1: To determine whether its service is satisfactory to its customers, a hotel surveyed 100 guests (40 females and 60 males). And the result is summarized in the table. Such a table is called a contin - enc table. —_—— __—-:fi- Def 1: Marginal probability is the probability of a single event without consideration of any other event. It’s also called simple probability. In the above example, if a guest is randomly selected from these 100 people, what is the probability that he! she is satisfied? Def 2: Conditional Probability is the probability that an event will occur given that another event has already occurred. If A and B are two events, then the conditional probability of A given B is written as P (A | B) and read as “the probability of A given that B has already occurr . “In general case P(A I B) at P(B | A) [n the above example, if we know the selected guest is a male, what is the probability that he is satisfied? How about the probability that a randomly selected guest is a male if we know that the person is satisfied? E); 2: To determine whether a consumer favor a new product, a company conducts a consumer survey in which a total of 425 customers are asked to try the product 5:“:[ and tell their opinions. The finding is listed in the table below. A person is l’ b {10" selected randoml from this group. in" __—-- —*m a). if you randomly select a person, what’s the probability that hez’she would not like {rm 9”“ij l brow.“ the pI‘OdUCt? we are lachtnj “F“ brfljl‘e "it/n40 b) If we are told that the person selected is a woman, then what is the probability that (Com/I bone] 56‘5““5‘ she does not like the product? jam!” 1133 been 91"! ct:- Ex 3: The table below shows the results of a survey in which researchers examined a person‘s gender and the dominant hand. _-_- ——_m 430 a) What is the probability that a randomly selected person is left—handed? b) What is the probability that a randomly selected person is left-handed given he a man? Section 4.6 Independent Versus Dependent Events 93f; Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. So A and B are independent events ifeither P(A | B) = P(A) or P(B| A) : P(B). Ifthey are not independent, they are called dependent events. Real-world Connections for Dependent Events If we wake up late, we will be late to office. Ifit rains, we use an umbrella. To determine if A and B are independent: 1. Calculate P(B) and P(BIA). 2. If they are equal, the events are independent. Otherwise, they are dependent. Ex 4: As in Ex 1, here is the contingenc table. Are events “Female” and “Satisfied" independent? How about “Male“ and “Not satisfied”? b) If we are told that the person selected is a woman, then what is the probability that (CMJ: howl E’s-“M’s she does not like the product? 9644.1: r [12% been S-Lie (it: Ex 3: The table below shows the results of a survey in which researchers examined a person’s gender and the dominant hand. _-_- __M 480 —m a) What is the probability that a randomly selected person is left-handed? b) What is the probability that a randomly selected person is left-handed given he a man? Section 4.6 Independent Versus Dependent Events 2913: Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. So A and B are independent events if either PM | B) = PM) or P(B] A): P(B). If they are not independent, they are called dependent events. Real-world Connections for Dependent Events If we wake up late, we will be late to office. If it rains, we use an umbrella. To determine if A and B are independent: 1. Calculate P(B) and P(B|A). 2. If they are equal, the events are independent. Otherwise, they are dependent. Ex 4: As in Ex 1, here is the contingency table. __— Are events “Female” and “Satisfied” independent? How about “Male” and “Not satisfied”? ...
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