Engineering Statistics Homework Notes

This document contains a comprehensive overview of various problems from an engineering statistics course, designed to enhance understanding of probability concepts and the application of statistical formulas across a range of scenarios. Each problem highlights a different application of statistics in engineering contexts.

Problem 1: Venn Diagrams
This problem involves the analysis of two events, A and B, using Venn diagrams to illustrate their relationships.

• Key Points:

  • Understand how to express relationships between events using unions (the combination of events) and intersections (the occurrence of both events simultaneously).

  • Use formulas for calculating probabilities of combined events, particularly focusing on how to derive and interpret probabilities from Venn diagrams.

Problem 2: Defect Probability
This scenario involves two inspectors, A and B, tasked with identifying defects in a batch of 10,000 items.

• Key Calculations:

  • Probability that inspector A found a defect:
    P(A) = \frac{724}{10000} = 0.0724

  • Probability that inspector B found a defect:
    P(B) = \frac{751}{10000} = 0.0751

  • Union of probabilities formula:
    P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
    This formula is crucial for understanding the total probability of finding defects by combining the independent probabilities of both inspectors.

Problem 3: Rolling Dice
The objective of this problem is to calculate the probability of rolling a sum greater than 3 but not exceeding 6 using two fair dice.

• Key Calculation:

  • Total possible outcomes for two dice: 36 outcomes

  • Favorable events (M): 12 events meet the condition of having a sum greater than 3 but ≤ 6.

  • Probability formula:
    P(3 < N1 + N2 \leq 6) = \frac{M}{36} = \frac{12}{36} = \frac{1}{3}

  • Evaluating all combinations of dice rolls is essential to attain the favorable outcomes and confirm the calculated probability.

Problem 4: Apparatus Failure Probability
This problem focuses on a system involving three independent electronic tubes, each with a failure probability of 0.04.

• Key Calculations:

  • Probability that one tube operates successfully:
    P(S_i) = 1 - 0.04 = 0.96

  • Probability that all tubes function:
    P(S) = P(S1) \times P(S2) \times P(S3) = 0.96^3 = 0.884736

  • Failure of the apparatus is calculated by the complement:
    P(F) = 1 - P(S) = 1 - 0.884736 \approx 0.115264
    This highlights the importance of evaluating individual failure rates and their cumulative effects on overall system reliability.

Problem 5: Computer Selection Events
This problem entails considering a selection of laptops and desktops from a total of 6 computers.

• Key Events:

  • Event A1: Selecting a laptop as the first setup.

  • Event A2: Selecting a laptop as the second setup.

• Calculation of probabilities based on total outcomes:

  • Total Outcomes: When picking pairs from the 6 computers, one must use combinations to determine the number of ways to select 2 computers from 6.

  • Probability of event where both selections are laptops is illustrated as just 1 favorable outcome out of a total of 15 possible outcomes:
    P(A1 \text{ and } A2) = \frac{1}{15}
    This problem elucidates concepts of combinatorics and probability in practical selections.

Problem 6: Extended Warranty Probability
This problem involves calculating the likelihood of choosing a model and purchasing an extended warranty using Bayes' Theorem, a critical concept in decision theory.

• Key Points:

  • The probability calculations for two models, A₁ and A₂, are as follows:
    P(B|A₁) = 0.3
    P(B|A₂) = 0.5

  • Using Bayes' theorem for posterior probability:
    P(A|B) = \frac{P(B|A)P(A)}{P(B|A₁)P(A₁) + P(B|A₂)P(A₂)}
    Mastery of Bayes' theorem allows for informed decision-making based on prior knowledge and likelihood estimates.

Problem 7: System of Components
This study focuses on evaluating a system comprising components connected in both series and parallel configurations.

• Key Definitions:

  • In parallel connections, the overall system functions as long as at least one component works.

  • In series connections, the system requires that both components work to function successfully.

• Calculation methods for the system's operational success: P(system works) = P(A1 \text{ works}) + P(A2 \text{ works}) - P(A1 \text{ and } A2 \text{ fail})

  • For components 3 and 4, which are connected in series, both must function properly to ensure overall system operation.
    This problem underlines the critical nature of evaluating system reliability through understanding independent and dependent component behaviors.