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In the first example, it is reasonable to assume that the class is independent, and each component probability is usually taken to be 0.5. In the second case, the assignmentof probabilities is somewhat more involved. For one thing, it is necessary to know the numbers of red and blue balls in each box before the composite trialbegins. When these are known, the usual assumptions and the properties of conditional probability suffice to assign probabilities. This approach of utilizingcomponent events is used tacitly in some of the examples in the unit on Conditional Probability.
When appropriate component events are determined, various Boolean combinations of these can be expressed as minterm expansions.
Four persons take one shot each at a target. Let E i be the event the i th shooter hits the target center. Let A 3 be the event exacty three hit the target. Then A 3 is the union of those minterms generated by the E i which have three places uncomplemented.
Usually we would be able to assume the E i form an independent class. If each is known, then all minterm probabilities can be calculated easily.
The following is a somewhat more complicated example of this type.
Ten race cars are involved in time trials to determine pole positions for an upcoming race. To qualify, they must post an average speed of 125 mph or more on atrial run. Let E i be the event the i th car makes qualifying speed. It seems reasonable to suppose the class is independent. If the respective probabilities for success are 0.90, 0.88, 0.93, 0.77, 0.85, 0.96,0.72, 0.83, 0.91, 0.84, what is the probability that k or more will qualify ( )?
The task of computing and adding the minterm probabilities by hand would be tedious, to say the least. However, we may use the function ckn, introduced in the unit on MATLAB and Independent Classes andillustrated in Example 4.4.2, to determine the desired probabilities quickly and easily.
>>P = [0.90, 0.88, 0.93, 0.77, 0.85, 0.96,0.72, 0.83, 0.91, 0.84];>>k = 6:10;>>PB = ckn(P,k)
PB = 0.9938 0.9628 0.8472 0.5756 0.2114 An alternate approach is considered in the treatment of random variables.
Many composite trials may be described as a sequence of success-failure trials. For each component trial in the sequence, the outcome is one of two kinds. One we designatea success and the other a failure . Examples abound: heads or tails in a sequence of coin flips, favor or disapprove of a proposition in a survey sample, and itemsfrom a production line meet or fail to meet specifications in a sequence of quality control checks. To represent the situation, we let E i be the event of a success on the i th component trial in the sequence. The event of a failure on the i th component trial is thus E i c .
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