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Chapter 13: Problem 17

During the repair of a large number of car engines it was found that part number 100 was changed in \(36 \%\) and part number 101 in \(42 \%\) of cases, and that both parts were changed in \(30 \%\) of cases. Is the replacement of part 100 connected with that of part \(101 ?\) Find the probability that in repairing an engine for which part 100 has been changed it will also be necessary to replace part 101 .

Short Answer

Expert verified
Yes, replacements are connected; the probability is 83.33%.

Step by step solution

01

Identify Given Probabilities

We need to identify the given probabilities: \(P(A)\), the probability that part 100 is changed, is 0.36 or 36%; \(P(B)\), the probability that part 101 is changed, is 0.42 or 42%; and \(P(A \cap B)\), the probability that both parts are changed, is 0.30 or 30%.
02

Calculate Conditional Probability

We are asked to find the probability that part 101 is replaced given that part 100 is replaced, denoted as \(P(B|A)\). This is calculated using the formula for conditional probability: \[P(B|A) = \frac{P(A \cap B)}{P(A)}.\]
03

Insert the Values into the Formula

Insert the given probabilities into the formula: \[P(B|A) = \frac{0.30}{0.36}.\]
04

Calculate the Conditional Probability

Perform the division: \[P(B|A) = \frac{0.30}{0.36} = 0.8333.\]
05

Interpret the Result

The probability that part 101 is replaced given that part 100 is replaced is approximately 0.83, or 83.33%. This value being significantly higher than just \(P(B)\) suggests a connection between the replacements of parts 100 and 101.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It is foundational to fields such as statistics, physics, finance, and many others. In probability theory, events are defined as outcomes or combinations of outcomes that might occur under specified conditions. The probability of an event is a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. The theory allows us to predict future events based on a model of chance or randomness.
  • **Probability**: This term refers to the likelihood of a particular event happening. In the context of the exercise, the probability of changing part 100, denoted as \(P(A)\), is 36%.
  • **Conditional Probability**: This is the probability of an event occurring given that another event has already occurred. The exercise focuses on this concept by asking if the replacement of part 100 affects the likelihood of also needing to replace part 101.
Understanding basic concepts like these helps to describe and analyze a range of random phenomena from everyday life.
Joint Probability
Joint probability refers to the likelihood of two events happening at the same time. It is a crucial concept in the realm of probability and statistics because it connects two different events, providing insights into their potential interactions or dependencies.

In the provided exercise, the joint probability \(P(A \cap B)\) is 30%, which represents the probability that both parts 100 and 101 are changed during a repair. By knowing the joint probability, one can analyze whether the occurrence of one event affects the occurrence of another. This is a crucial aspect when investigating potential dependencies or relationships between events.
  • **Calculation of Joint Probability**: In general, joint probability is calculated when you multiply the probabilities of two independent events, but when events are dependent, other methods, such as conditional probability, are used.
  • **Importance**: Assessing joint probabilities helps in various practical scenarios, like detecting dependency and association between different occurrences.
Mathematical Statistics
Mathematical statistics involves the application of statistical theory where the focus is on deriving new statistical methods through mathematical principles. These methods are used to analyze real-world data and provide insights or forecasts.

The exercise involves doing a statistical analysis to estimate the correlation between two events鈥攔eplacing parts 100 and 101 in car engines. By calculating probabilities and assessing their interactions, one can make data-driven decisions or predictions.
  • **Analysis Tools**: Using conditional probabilities and joint probabilities, as seen in this problem, supports effective decision-making in engineering or maintenance contexts.
  • **Application**: Practicing statistical methods allows engineers and statisticians to predict maintenance needs and improve operational efficiency.
Statistical methods enable us to extract logical conclusions about the dependencies between variables using data from observations and experiments.
Problem Solving Steps
Approaching complex problems is made easier with a structured method called problem-solving steps. This approach helps unravel challenging questions by breaking them into manageable parts.

Let's see how the provided exercise is resolved with clear steps:
  • **Step 1: Identifying Probabilities**: Start by determining the key probabilities involved. It's essential to clearly define events and their likelihood, like parts being replaced.
  • **Step 2: Calculate Conditional Probability**: Use the formula \(P(B|A) = \frac{P(A \cap B)}{P(A)}\), which helps find the probability of one event given the occurrence of another.
  • **Step 3: Substitute Known Values**: Insert the given probabilities into the formula to find \(P(B|A)\).
  • **Step 4: Perform Calculations**: Execute the arithmetic operations to determine the result.
  • **Step 5: Interpreting Results**: Analyze the calculated probability to understand the relationship or dependence between the events, i.e., parts replacements.
By following these steps, complex problems in probability theory can be solved methodically and with confidence, leading to insightful conclusions.

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Most popular questions from this chapter

At the 18 th hole of a golf course the probability that a golfer will score a par four is \(0.55\), the probability of one under is \(0.17\), of two under is \(0.03\), of one over is \(0.2\) and of two over is \(0.05\). Plot the (cumulative) distribution function.

The mean times for completion of tasks \(\mathrm{A}\) and \(\mathrm{B}\) are four and six hours respectively. A particular project involves three tasks of type \(\mathrm{A}\) and two of type \(\mathrm{B}\), all to be performed in succession. What is the expected time for completion of the project? Also, if the standard deviations for \(\mathrm{A}\) and \(\mathrm{B}\) are one and two hours respectively, and if all project times are independent, what is the standard deviation of the completion time?

Two people are separately attempting to succeed at a particular task, and each will continue attempting until success is achieved. The probability of success of each attempt for person \(\mathrm{A}\) is \(p\), and that for person B is \(q\), all attempts being independent. What is the probability that person B will achieve success with no more attempts than person A does? $$ \left(\text { Hint: } \sum_{i=0}^{n-1} x^{i}=\frac{1-x^{n}}{1-x}\right) $$

Part of an electrical circuit consists of three elements \(K, L\) and \(M\) in series. Probabilities of failure for elements \(K\) and \(M\) during operating time \(t\) are \(0.1\) and \(0.2\) respectively. Element \(L\) itself consists of three sub-elements \(L_{1}, L_{2}\) and \(L_{3}\) in parallel, with failure probabilities \(0.4,0.7\) and \(0.5\) respectively, during the same operating time \(t\). Find the probability of failure of the circuit during time \(t\), assuming that all failures of elements are independent.

An inspection of twelve specimens of material from inside a reactor vessel revealed the following percentages of impurities: $$ \begin{aligned} &2.3,1.9,2.1,2.8,2.3,3.6,1.4,1.8,2.1,3.2 \\ &2.0,1.9 \end{aligned} $$ Find (a) the sample average and both versions of the sample standard deviation, (b) the sample median and range.
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