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Chapter 16: Problem 17

The reliability of an engine working to specification is \(0.85\). Two engines are tested at random. Find the probability that at least one of the engines is working to specification.

Short Answer

Expert verified
0.9775

Step by step solution

01

Understand the Problem

We need to find the probability that at least one of the two engines is working to specification.
02

Identify Key Probabilities

The probability that one engine works to specification is given as 0.85. Let鈥檚 denote this as P(W) = 0.85. Consequently, the probability that one engine does not work to specification is P(NW) = 1 - P(W) = 0.15.
03

Calculate Probability That Both Engines Are Not Working

The probability that both engines are not working is found by multiplying the probabilities of each engine failing: P(Both NW) = P(NW) * P(NW) = 0.15 * 0.15 = 0.0225.
04

Calculate Probability That At Least One Engine Is Working

To find the probability that at least one engine is working, subtract the probability that both engines are not working from 1: P(At least one W) = 1 - P(Both NW) = 1 - 0.0225 = 0.9775.
05

Conclusion

The probability that at least one of the two engines is working to specification is 0.9775.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reliability
Reliability is an important concept in probability theory and statistics.
It refers to the likelihood that a system or component operates without failure under given conditions for a specified period of time.
In this exercise, the reliability of an engine working to specification is given as 0.85. This means there is an 85% chance that any randomly chosen engine will work properly.
To put it simply, it's a measure of how consistently the engine performs as expected.
  • High reliability indicates greater consistency and fewer failures.
  • Low reliability means more frequent failures.
Understanding reliability allows us to predict performance and plan maintenance.
Complement Rule
The complement rule is a fundamental principle in probability theory.
It states that the probability of an event happening is equal to 1 minus the probability of it not happening. In mathematical terms, if \(P(A)\) is the probability of event A occurring, then \(P(A^c) = 1 - P(A)\) is the probability of event A not occurring.
In this exercise:
  • The probability of one engine working to specification is 0.85 (\(P(W)\)).
  • The probability of one engine not working to specification is 0.15 (\(P(NW) = 1 - P(W)\)).
We used the complement rule to find the probability that both engines fail. From there, we computed the probability that at least one engine works by subtracting this failure probability from 1.
Independent Events
Independent events are events whose outcomes do not affect each other.
In other words, the occurrence of one event does not influence the occurrence of another.
In this exercise, the functioning of each engine is an independent event because whether one engine works does not impact the other.
To compute combined probabilities involving independent events, we multiply their individual probabilities:
  • The probability of both engines failing is the product of each engine failing (\(0.15 * 0.15 = 0.0225\)).
  • If we wanted to find the probability of both engines working, we would multiply their success probabilities (\(0.85 * 0.85\)).
Understanding independence helps us to simplify complex problems by breaking them down into simpler probability calculations.

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Most popular questions from this chapter

A calculator manufacturer finds that \(5 \%\) of the calculators do not have a memory button, \(3 \%\) do not have a log button and \(1 \%\) do not have log and memory buttons. For a randomly selected calculator, find the probability that the calculator a does not have a memory button or a log button b does have a memory button and a log button.

Determine the mean and standard deviation for the following data: a \(5,7,12,15,37\) b \(105,107,112,115,137\) c \(0.5,0.7,1.2,1.5,3.7\) d \(53,73,123,153,373\) What do you notice about your results? [For questions 3 and 4: \(\bar{x}\) and \(\bar{y}\) represent \(x\) mean and \(y\) mean respectively. \(s_{x}\) and \(s_{y}\) represent the standard deviations for \(x\) and \(y\) respectively. Also \(j=1,2, \ldots, n\).]

Determine the probability of having more boys than girls in a family of 7 children. Assume the probability of having a boy is \(0.48\).

Assume the probability of having a girl or a boy is equally likely. Determine the expected number of girls in a family which has three children. Comment on your result.

The heights of 500 students follow a normal distribution with a mean of \(1.70 \mathrm{~m}\) and a standard deviation of \(0.15 \mathrm{~m}\). Determine the number of students who have a height a greater than \(1.80 \mathrm{~m}\) b less than \(1.50 \mathrm{~m}\) c between \(1.65 \mathrm{~m}\) and \(1.85 \mathrm{~m}\).
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