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Chapter 2: Problem 52

1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the extra remaining pump is 鈥� # A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the extra remaining pump is now more likely to fail than was originally the case. That is, \(r=P(\\# 2\) fails \(\mid\) # 1 fails \()>P(\\# 2\) fails \()=q\). If at least one pump fails by the end of the pump design life in \(7 \%\) of all systems and both pumps fail during that period in only \(1 \%\), what is the probability that pump #1 will fail during the pump design life?

Short Answer

Expert verified
The probability that pump #1 will fail is 0.04 or 4%.

Step by step solution

01

Define the Problem

We need to determine the probability that pump #1 will fail during the pump's design life, given the probabilities that at least one pump fails are \(7\%\), and both pumps fail are \(1\%\).
02

Translate Probabilities into Events

Let \( P(A) \) be the probability that pump #1 fails, \( P(B) \) be the probability that pump #2 fails. \( P(\text{At least one fails}) = 0.07 \) and \( P(A \cap B) = 0.01 \).
03

Use the Formula for At Least One Event

The probability of at least one pump failing can be expressed as: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Given that \( P(A \cup B) = 0.07 \) and \( P(A \cap B) = 0.01 \), we substitute these values into the equation.
04

Express the Relationship in Terms of Probabilities

We know the total probability of the system failing: \[ 0.07 = P(A) + P(B) - 0.01 \].
05

Assume Symmetry for Pumps

Since the problem does not specify otherwise, assume initially that each pump has an equal chance of failing individually under normal circumstances: \( P(A) \approx P(B) \).
06

Calculate Individual Probabilities

Substitute \( P(A) = P(B) \) into the equation:\[ 0.07 = 2P(A) - 0.01 \]Solving for \(P(A)\):\[ 2P(A) = 0.07 + 0.01 = 0.08 \]\[ P(A) = 0.04 \].
07

Conclusion

Since both pumps were assumed identical initially, and we're seeking \( P(A) \), the probability that pump #1 will fail during the pump design life is \( 0.04 \).

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Key Concepts

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

Conditional Probability
Conditional probability is the likelihood of an event occurring, given that another event has already happened. It helps us understand the relationship between two dependent events. In our exercise, the failure of one pump influences the likelihood of the other failing. Mathematically, conditional probability is denoted as \( P(B \mid A) \), representing the probability of event \( B \) occurring given that \( A \) has occurred.

To calculate conditional probability, we use the formula:
  • \( P(B \mid A) = \frac{P(A \cap B)}{P(A)} \)
In the context of the pump system, this relationship is pivotal in predicting the system's overall performance when one pump fails. Given the information that both pumps failing is only 1%, we need to understand how the failure of one pump impacts the probability outcomes for the system.
Failure Analysis
Failure analysis involves examining systems to determine how they might fail and the probabilities associated with these failures. In the pump system exercise, failure analysis helps identify the system weakness and the potential impact of strain on the second pump when the first one fails.

Key elements of failure analysis include:
  • Identifying all components that might fail (here, the pumps)
  • Determining interdependencies (the increased chance of the second pump failing after the first)
  • Calculating overall system failure probabilities (such as the 7% probability of at least one pump failure)
Having these insights allows us to compute more accurate probabilities and improve system designs by mitigating identified risks. In this way, analyzing failures using statistical probabilities ensures better preparedness and potentially increased reliability in practical scenarios.
Mathematical Statistics
Mathematical statistics is a branch of mathematics that uses probability theory to analyze data and make conclusions about real-world scenarios. In the pump example, mathematical statistics help us derive important values like the probabilities of pump failures.

This branch involves:
  • Using probability formulas, such as \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), to structure problem-solving approaches
  • Applying assumptions like symmetry (equal probability of each pump failing) to simplify calculations
  • Deriving individual probabilities through structured formulas, helping in concluding that \( P(A) = 0.04 \), where \( P(A) \) is the probability of pump #1鈥檚 failure
Mathematical statistics provides a foundation not only for solving such problems but also for making informed decisions based on data. By carefully breaking down problems and applying statistical techniques, we can more reliably predict outcomes and inform designs or decisions, in this case, predicting pump reliability and refining engineering strategies.

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Most popular questions from this chapter

A certain shop repairs both audio and video components. Let \(A\) denote the event that the next component brought in for repair is an audio component, and let \(B\) be the event that the next component is a compact disc player (so the event \(B\) is contained in \(A\) ). Suppose that \(P(A)=.6\) and \(P(B)=.05\). What is \(P(B \mid A)\) ?

Four graduating seniors, \(A, B, C\), and \(D\), have been scheduled for job interviews at 10 a.m. on Friday, January 13, at Random Sampling, Inc. The personnel manager has scheduled the four for interview rooms \(1,2,3\), and 4 , respectively. Unaware of this, the manager's secretary assigns them to the four rooms in a completely random fashion (what else!). What is the probability that a. All four end up in the correct rooms? b. None of the four ends up in the correct room?

A personnel manager is to interview four candidates for a job. These are ranked \(1,2,3\), and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order \(3,4,1,2\) generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3,4 , 2,1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first \(s\) candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with \(s=2\), the order \(3,4,1\), 2 would result in the best being hired, whereas the order \(3,1,2,4\) would not. Of the four possible \(s\) values \((0,1,2\), and 3\()\), which one maximizes \(P\) (best is hired)? [Hint: Write out the 24 equally likely interview orderings: \(s=0\) means that the first candidate is automatically hired.]

Suppose a single gene determines whether the coloring of a certain animal is dark or light. The coloring will be dark if the genotype is either \(A A\) or \(A a\) and will be light only if the genotype is \(a a\) (so \(A\) is dominant and \(a\) is recessive). Consider two parents with genotypes \(A a\) and \(A A\). The first contributes \(A\) to an offspring with probability \(1 / 2\) and \(a\) with probability \(1 / 2\), whereas the second contributes \(A\) for sure. The resulting offspring will be either \(A A\) or \(A a\), and therefore will be dark colored. Assume that this child then mates with an \(A a\) animal to produce a grandchild with dark coloring. In light of this information, what is the probability that the first-generation offspring has the \(A a\) genotype (is heterozygous)? [Hint: Construct an appropriate tree diagram.]

One method used to distinguish between granitic \((G)\) and basaltic \((B)\) rocks is to examine a portion of the infrared spectrum of the sun's energy reflected from the rock surface. Let \(R_{1}, R_{2}\), and \(R_{3}\) denote measured spectrum intensities at three different wavelengths; typically, for granite \(R_{1}P\) (basalt \(\mid R_{1}
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