There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?
The answer islocalid="1646910197664"
Let denotes the number of non functional components and let denotes the event. if no two nonfunctional components are to be constructive, then the space between the functional components must each contain at most one non functional components.
,(From Bayes theorem)
The answer is localid="1646910184882"
Here is another way to obtain a set of recursive equations for determining , the probability that there is a string of consecutive heads in a sequence of flips of a fair coin that comes up heads with probability :
(a) Argue that for , there will be a string of consecutive heads if either
1. there is a string of consecutive heads within the first flips, or
2. there is no string of consecutive heads within the first flips, flip is a tail, and flips are all heads.
(b) Using the preceding, relate . Starting with , the recursion can be used to obtain , then , and so on, up to .
Suppose that it takes at least votes from a - member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is whereas the probability that the juror votes an innocent person guilty is If each juror acts independently and if percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted?
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