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?20
Application of Combinations to Probability Problems.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:21
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?DiameterLengthToo ThinOKToo ThickSumToo Short103518OK389004942Too Long2251340Sum5092822100022
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?