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Chapter 4: Problem 115

The failure rate for a guided missile control system is 1 in \(1000 .\) Suppose that a duplicate, but completely independent, control system is installed in each missile so that, if the first fails, the second can take over. The reliability of a missile is the probability that it does not fail. What is the reliability of the modified missile?

Short Answer

Expert verified
Answer: The reliability of the modified missile is 999,999/1,000,000.

Step by step solution

01

Calculate the probability of failure of each control system

Since the failure rate for each control system is 1 in 1000, the probability of a single control system failing is: \(P(Failure)=\frac{1}{1000}\)
02

Calculate the probability of both systems failing

Since the two control systems are independent, the probability of both systems failing can be found by multiplying their individual failure probabilities: \(P(Both\ Failures)=P(Failure)\cdot P(Failure)=\frac{1}{1000}\cdot \frac{1}{1000}=\frac{1}{1000000}\)
03

Calculate the reliability of the modified missile

To find the reliability of the modified missile, we can calculate the complementary event of both systems failing, which represents the probability that at least one of the systems does not fail (the modified missile does not fail). This can be done using the following formula: \(P(Reliability) = 1 - P(Both\ Failures) = 1 - \frac{1}{1000000} = \frac{999999}{1000000}\) The reliability of the modified missile is \(\frac{999999}{1000000}\).

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Most popular questions from this chapter

During the inaugural season of Major League Soccer in the United States, the medical teams documented 256 injuries that caused a loss of participation time to the player. The results of this investigation, reported in The American Journal of Sports Medicine, is shown in the table. \({ }^{3}\) If one individual is drawn at random from this group of 256 soccer players, find the following probabilities: a. \(P(A)\) b. \(P(G)\) c. \(P(A \cap G)\) d. \(P(G \mid A)\) e. \(P(G \mid B)\) f. \(P(G \mid C)\) g. \(P(C \mid P)\) h. \(P\left(B^{c}\right)\)

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A particular football team is known to run \(30 \%\) of its plays to the left and \(70 \%\) to the right. A linebacker on an opposing team notes that the right guard shifts his stance most of the time \((80 \%)\) when plays go to the right and that he uses a balanced stance the remainder of the time. When plays go to the left, the guard takes a balanced stance \(90 \%\) of the time and the shift stance the remaining \(10 \%\). On a particular play, the linebacker notes that the guard takes a balanced stance. a. What is the probability that the play will go to the left? b. What is the probability that the play will go to the right? c. If you were the linebacker, which direction would you prepare to defend if you saw the balanced stance?

Only \(40 \%\) of all people in a community favor the development of a mass transit system. If four citizens are selected at random from the community, what is the probability that all four favor the mass transit system? That none favors the mass transit system?

Use event relationships to fill in the blanks in the table below. $$ \begin{array}{|c|c|l|c|c|c|} \hline \boldsymbol{P}(\boldsymbol{A}) & \boldsymbol{P}(\boldsymbol{B}) & \text { Conditions for Events } \boldsymbol{A} \text { and } \boldsymbol{B} & \boldsymbol{P}(\boldsymbol{A} \cap \boldsymbol{B}) & \boldsymbol{P}(\boldsymbol{A} \cup \boldsymbol{B}) & \boldsymbol{P}(\boldsymbol{A} \mid \boldsymbol{B}) \\ \hline .3 & .4 & \text { Mutually exclusive } & & & \\ \hline .3 & .4 & \text { Independent } & & & \\ \hline .1 & .5 & & & & .1 \\ \hline .2 & .5 & & 0 & & \\ \hline \end{array} $$
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