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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(AB)=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 dietoss 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 4.74.9, we can solve the following problems.
Ex 6: A woman ’s clothing store owner buys from three companies: A, B and C. The
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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 lefthanded 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. Realworld 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 SLie (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 lefthanded? b) What is the probability that a randomly selected person is lefthanded 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. Realworld 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(BA). 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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