Example 2.11.Continuing from Example 2.10, assume instead of a board we are interested in thenumber of possiblecommittees, where each member has equal power. How many such committees are possible?
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Application of Combinations to Probability Problems
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Example 2.13.A lot of 100 articles contains 10 defective. If a sample of 5 articles is chose at random,what is the probability that it contains 3 defective?
Example 2.14.A player serving at tennis is only allowed one fault. At a double fault, the sever losesa point and the other player gains a point. Given the following information:
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What is the probability that the server loses a point,i.e.,what is the probability that both servicesresult in faults?Example 2.15.Referring to Example 2.6. Given that the length of a rod is too long, what is theprobability that the diameter is OK?
Diameter
Length
Too Thin
OK
Too Thick
Sum
Too Short
10
3
5
18
OK
38
900
4
942
Too Long
2
25
13
40
Sum
50
928
22
1000
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2.4.1Independent EventsWhen the given occurrence of one event does not influence the probability of a potential outcome ofanother event, then the two events are said to be independent.Two eventsAandBareindependentif the probability of each remains the same, whether or notthe other has occurred.AandBare independent ifP(A)6= 0 andP(B)6= 0, thenP(B|A) =P(B)⇐⇒P(A|B) =P(B)IfP(A) = 0 orP(B) = 0, then the two events are independent.More generally, the eventsA1, ...Anare independent if for eachAiand each collectionAj1, ..., Ajmofevents withP(Aj1∩ · · · ∩Ajm)6= 0P(Ai|Aj1, ..., Ajm) =P(Ai)As a consequence of independence, the rule of multiplication then saysP(A∩B) =P(A|B)P(B) =P(A)P(B)In general,Pk\i=1Ai!=kYi=1P(Ai)k≥1Example 2.16.Of the microprocessors manufactured by a certain process, 20% of them are defec-tive. Assume they function independently. Five microprocessors are chosen at random. What is theprobability that they will all work?