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Chapter 4: Q. 4.33 (page 172)
Repeat Theoretical Exercise 4.32, this time assuming that withdrawn chips are not replaced before the next selection.
Probability mass function does not exist
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Consider n independent trials, each of which results in one of the outcomes with respective probabilities Show that if all the are small, then the probability that no trial outcome occurs more than once is approximately equal to.
A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on the day, then each of his examiners will pass him, independently of one another, with probability, whereas if he has an off day, this probability will be reduced to. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on the day, should he request an examination withexaminers or withexaminers?
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?
There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1 − p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p, Ci, Ri, i = 1, 2, should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?
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 .
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