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Chapter 4: Q. 4.32 (page 172)
A jar contains chips. Suppose that a boy successively draws a chip from the jar, each time replacing the one drawn before drawing another. The process continues until the boy draws a chip that he has previously drawn. Let denote the number of draws, and compute its probability mass function.
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A communications channel transmits the digits and However, due to static, the digit transmitted is incorrectly received with probability Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of and 11111 instead of If the receiver of the message uses 鈥渕ajority鈥� decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?
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 .
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 鈭� 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?
A satellite system consists of components and functions on any given day if at least of the n components function on that day. On a rainy day, each of the components independently functions with probability whereas, on a dry day, each independently functions with probability . If the probability of rain tomorrow is what is the probability that the satellite system will function?
A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.
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