APPROXIMATIONS TO LARGE PROBABILITIES OF ALL SUCCESSES FOR GENERAL CASE AND SOME OPERATIONS RESEARCH IMPLICATIONS
SOUTHERN METHODIST UNIV DALLAS TX DEPTOF STATISTICAL SCIENCE
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There are many cases where an overall effort is successful if and only if the efforts or events of a sequence are all successful. Often, the principal interest is in cases where overall success has a large probability say, at least .8. Suppose there are n efforts in the sequence and that p sub i s sub i-1 is the probability that the i-th effort is a success given that preceding efforts 1, ..., i-1 are successes i1, ..., n where s sub o denotes no conditions. An approximate value, also sharp upper and lower bounds, are developed for the probability that all n events are successes. This is done for various levels of generality, including a form of complete generality. These results depend only on n, the generality level, and the arithmetic average of the p sub i s sub i-1. They are useful when the probability of all successes is at least .8 then the approximate value is near both bounds. The necessity of only considering the arithmetic average of the p sub i s sub i-1 , rather than their product, sometimes can be useful in analyses of an operations research nature including reliability situations.
- Operations Research