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p93_exercises-3_7

# p93_exercises-3_7 - What is the expected number of...

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Unformatted text preview: What is the expected number of opponents contested in a 3;. l e? [I What is the probability that a player contests four or more Kopponents in a game? i, What is the expected number of game plays until a player ,contests four or more opponents? 1;: . Heart failure is due to either natural occurrences (87%) ide factors (13%). Outside factors are related to induced 1.068 or foreign objects. Natural occurrences are caused .w - 'al blockage, disease, and infection. Assume that causes failure between individuals are independent. = What is the probability that the ﬁrst patient with heart fail- I are that enters the emergency room has the condition due ’ to outside factors? j What is the probability that third patient with heart failure ‘ that enters the emergency room is the ﬁrst one due to out- side factors? . 3 What is the mean number of heart failure patients with the ‘ condition due to natural causes that enter the emergency room before the ﬁrst patient with heart failure from out- , 'side factors? 3—8 HYPERGEOMETRIC DISTRIBUTION 93 (c) What is the mean number of days until all eight computers fail in the same day? 3—92. Assume that 20 parts are checked each hour and that X denotes the number of parts in the sample of 20 that require rework. Parts are assumed to be independent with respect to rework. (a) If the percentage of parts that require rework remains at 1%, what is the probability that hour 10 is the ﬁrst sample at whichX exceeds 1? (b) If the rework percentage increases to 4%, what is the probability that hour 10 is the ﬁrst sample at which X exceeds 1? (c) If the rework percentage increases to 4%, what is the expected number of hours until X exceeds 1? 3—93. A fault-tolerant system that processes transactions for a ﬁnancial services ﬁrm uses three separate computers. If the operating computer fails, one of the two spares can be imme- diately switched online. After the second computer fails, the last computer can be immediately switched online. Assume that the probability of a failure during any transaction is 10‘8 and that the transactions can be considered to be independent 1‘ A computer system uses passwords constructed from g26 letters (a—z) or 10 integers (0—9). Suppose there are 1"” users on the system with unique passwords. A hacker nomly selects (with replacement) passwords from the po- ’T'Il set. .There are 9900 users with unique six-character pass- words on the system and the hacker randomly selects six— . character passwords. What is the mean and standard devi- . ation of the number of attempts before the hacker selects a ' user password? Suppose there are 100 users with unique three-character I passwords on the system and the hacker randomly selects three-character passwords. What is the mean and standard deviation of the number of attempts before the hacker selects a user password? Comment on the security differences between six- and ;- three-character passwords. events. (a) What is the mean number of transactions before all com- puters have failed? (b) What is the variance of the number of transactions before all computers have failed? 3—94. In the process of meiosis, a single parent diploid cell, goes through eight different phases. However, only 60% of the processes pass the ﬁrst six phases and only 40% pass all eight. Assume the results from each phase are independent. (a) If the probability of a successful pass of each one of the ﬁrst six phases is constant, what is the probability of a suc- cessful pass of a single one of these phases? (b) If the probability of a successful pass of each one of the last two phases is constant, what is the probability of a successful pass of a single one of these phases? 3—95. Show that the probability density function of a nega- tive binomial random variable equals the probability density function of a geometric random variable when r = 1. Show that the formulas for the mean and variance of a negative bi- nomial random variable equal the corresponding results for a geometric random variable when r = 1. '15 1. A trading company has eight computers that it uses to 5w on the New York Stock Exchange (NYSE). The proba- ‘ty of a computer failing in a day is 0.005, and the comput- '. fail independently. Computers are repaired in the evening it each day is an independent trial. What is the probability that all eight computers fail in a day? What is the mean number of days until a speciﬁc com- puter fails? 3—96. Derive the expression for the variance of a geometric random variable with parameter p. (Formulas for inﬁnite se- ries are required.) 8 HYPERGEOMETRIC DISTRIBUTION In Example 3-8, a day’s production of 850 manufactured parts contains 50 parts that do not conform to customer requirements. Two parts are selected at random, without replacement from the day’s production. That is, selected units are not replaced before the next selection is ...
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