Probability, Statistics and Reliability

Introduction and Context

Machine technology depends on decisions made under uncertainty. A manufactured dimension varies from part to part, a cutting tool may fail before its expected life, and a maintenance engineer must estimate future breakdowns from incomplete records. Probability provides a language for uncertainty; statistics provides methods for learning from data; reliability engineering applies both to the dependable performance of machines and components.

In quality and maintenance, these disciplines support questions such as:

• How likely is a dimension to fall outside specification?
• How many components should be sampled from a production lot?
• Is a new cutting-fluid formulation improving tool life?
• Does spindle speed significantly affect surface roughness?
• What is the probability that a machine operates for a specified time without failure?
• When should preventive maintenance be scheduled?

The central principle is that statistical conclusions are useful only when the data-generation process, assumptions, and consequences of error are understood.

Learning Objectives

By the end of this chapter, the student should be able to:

1. Explain probability concepts and select suitable probability distributions.
2. Calculate and interpret descriptive statistics and sampling estimates.
3. Construct confidence intervals and perform hypothesis tests.
4. Interpret correlation and fit simple regression models.
5. Explain basic design-of-experiments principles.
6. Apply statistical process control to manufacturing data.
7. Calculate reliability, failure-rate, and maintainability measures.
8. Use statistical evidence responsibly in quality and maintenance decisions.

1. Foundations of Probability

A sample space, denoted by S, is the set of all possible outcomes of an experiment. An event is a subset of S. The probability P(A) of event A satisfies 0 ≤ P(A) ≤ 1, with P(S) = 1.

For equally likely outcomes,

For any events A and B, the addition rule is

where A ∪ B means “A or B” and A ∩ B means “A and B.” If A and B are mutually exclusive, P(A ∩ B) = 0.

The conditional probability of A given B is

If A and B are independent, then P(A | B) = P(A), and

The law of total probability is useful when an outcome can arise from several mutually exclusive causes B₁, B₂, …, Bₖ:

Bayes’ theorem reverses conditional probabilities:

It is important in fault diagnosis. A positive inspection result does not necessarily mean that a fault is present; the base rate of the fault and the test accuracy must also be considered.

Worked Example 1: Inspection Evidence

Suppose 2% of bearings in a population have a defect. An inspection system detects a defective bearing with probability 0.95 and incorrectly flags a good bearing with probability 0.03. The probability that a flagged bearing is actually defective is

Thus, only about 39.3% of flagged bearings are defective. Additional inspection may be justified before rejecting an expensive batch.

2. Random Variables and Probability Distributions

A random variable assigns a numerical value to each outcome. A discrete random variable has countable values, while a continuous random variable can take values throughout an interval. Its expected value or mean is the long-run average:

where f(x) is a probability density and the integral is taken over all possible values. Variance measures spread:

where μ = E(X), and the standard deviation is σ = √Var(X). Variance has squared units; standard deviation has the same units as X.

Important Distributions

The binomial model requires a fixed number n of trials, two outcomes per trial, constant probability p, and independence. The Poisson model counts events in a fixed time, length, area, or volume when events occur independently at a constant average rate λ.

For a binomial random variable X,

where C(n,x) = n! ÷ [x!(n−x)!]. For a Poisson variable,

The normal distribution has density

The standardized value is

The standard normal model is often used to estimate the proportion of dimensions within tolerance, provided the measurement distribution is approximately normal and μ and σ are suitable estimates.

Worked Example 2: Poisson Defects

A machining cell averages 1.5 surface defects per panel. If defects follow a Poisson distribution, the probability of exactly two defects on a panel is

The result is approximately 25.1%. This model should not be used automatically if defects cluster, interact, or arise from changing process conditions.

3. Sampling and Descriptive Statistics

A population contains all units of interest; a sample is the observed subset. A probability sample gives units a known basis for selection and helps avoid selection bias. Common designs include simple random, systematic, stratified, and cluster sampling.

For observations x₁, x₂, …, xₙ, the sample mean is

The sample variance and sample standard deviation are

The divisor n − 1 provides an unbiased estimator of population variance under random sampling from a population with finite variance. The median is the middle ordered value, and the interquartile range is Q₃ − Q₁. The coefficient of variation is s ÷ x̄, usually expressed as a percentage when x̄ is positive; it compares relative variability but is inappropriate when the mean is near zero.

The standard error of the sample mean, for independent observations, is

When σ is unknown, s replaces σ. If sampling without replacement from a finite population of size N, a finite-population correction may be used:

This correction matters when the sample is a substantial fraction of the population.

4. Estimation and Hypothesis Testing

A point estimate gives one value; an interval estimate gives a range of plausible values. For a mean with known population standard deviation σ, a 100(1−α)% confidence interval is

where z₍α/2₎ is the standard-normal critical value. If σ is unknown and the population is approximately normal, use the t distribution:

The confidence level describes the long-run performance of the interval procedure, not the probability that a particular fixed parameter randomly lies inside an already calculated interval.

For a population proportion p, the large-sample interval is approximately

provided the expected counts of successes and failures are sufficiently large. Exact or improved methods may be preferable for small samples or extreme proportions.

Hypothesis Testing

A test begins with a null hypothesis H₀ and an alternative hypothesis H₁. The significance level α is the probability of rejecting H₀ when H₀ is true. A Type I error is a false rejection; a Type II error is failure to reject a false H₀. The p-value is the probability, assuming H₀ is true, of obtaining a result at least as extreme as the observed result under the specified test statistic.

A two-sided one-sample mean test with known σ uses

where μ₀ is the mean specified by H₀. When σ is unknown, use

with n−1 degrees of freedom. The test requires independent observations and, for small samples, an approximately normal population.

Statistical significance is not the same as practical significance. A very small improvement may be statistically detectable with a large sample but economically irrelevant. Conversely, an important improvement may fail to achieve significance with inadequate sample size.

Worked Example 3: Mean Diameter

A shaft specification requires a target mean diameter of 25.00 mm. A random sample of 16 shafts has x̄ = 25.06 mm and s = 0.08 mm. Test H₀: μ = 25.00 mm against H₁: μ ≠ 25.00 mm at α = 0.05.

With 15 degrees of freedom, the two-sided 5% critical value is approximately 2.131. Since |3.00| > 2.131, reject H₀. The evidence indicates that the process mean differs from 25.00 mm. Engineers should investigate centring, measurement bias, and the practical effect on assembly rather than simply label the process defective.

5. Correlation and Regression

The Pearson correlation coefficient measures linear association between paired variables X and Y:

Its value lies between −1 and +1. A positive value indicates that larger X values tend to accompany larger Y values; a negative value indicates the reverse. Correlation is unit-free but is sensitive to outliers and does not establish causation.

Simple linear regression models the conditional mean of Y:

where

and b₀ = ȳ − b₁x̄. The slope b₁ has units of Y per unit X. The coefficient of determination R² describes the proportion of observed variation in Y explained by the fitted linear model in the sample; in simple regression, R² = r².

Regression inference assumes suitable linearity, independent errors, constant error variance, and approximately normal errors for small-sample interval or test procedures. Residual plots should be examined. Extrapolation beyond the observed range is particularly risky.

6. Design of Experiments

Design of experiments (DOE) changes controllable factors systematically to determine their effects on a response. Compared with changing one factor at a time, factorial designs can reveal interactions.

A factor is an input such as cutting speed; a level is a selected setting such as 80 or 120 m/min; a response is an output such as tool wear. An interaction occurs when the effect of one factor depends on the level of another.

Important principles are:

• Randomization: run order is randomized to reduce bias from time-related effects.
• Replication: repeated experimental runs estimate random error and improve precision.
• Blocking: known nuisance variation, such as machine or operator, is grouped into blocks.
• Control of conditions: measurement methods and relevant environmental conditions are standardized.

In a two-factor factorial experiment, an additive model may be written

where μ is the overall mean, αᵢ and βⱼ are main effects, (αβ)ᵢⱼ is the interaction, and εᵢⱼₖ is random error. Analysis of variance (ANOVA) compares variation attributable to model terms with residual variation using an F statistic:

ANOVA assumptions include independent errors, approximately normal errors, and equal variance across treatment combinations. Transformations or nonparametric methods may be needed when these assumptions are seriously violated.

7. Statistical Process Control

Statistical process control (SPC) distinguishes common-cause variation, inherent in a stable process, from special-cause variation produced by identifiable changes such as tool damage, incorrect setup, or sensor failure. Control limits are calculated from process data; specification limits are imposed by design or customer requirements. They are not interchangeable.

For individual measurements from a stable approximately normal process, approximate three-sigma limits are

where UCL, CL, and LCL are the upper control limit, centre line, and lower control limit. In practice, μ and σ are estimated, and the appropriate chart constants depend on the sampling plan.

For sample means of size n, if process parameters are known,

An x̄ chart monitors process centring; an R chart monitors within-subgroup range; an S chart monitors within-subgroup standard deviation. For attribute data, p charts monitor fraction nonconforming and c or u charts monitor counts of defects under suitable assumptions.

A process capability index, when the process is stable and approximately normal, is

where USL and LSL are upper and lower specification limits. The centred capability index is

Capability indices are meaningful only after stability has been established. A stable but poorly centred process may have a low Cₚₖ even when Cₚ is large.

8. Reliability Engineering

Reliability R(t) is the probability that an item performs its required function for at least time t under stated operating conditions. The failure distribution function is F(t) = 1 − R(t). The probability density of failure time is f(t), and the hazard or failure-rate function is

The hazard has units of inverse time. It is a conditional instantaneous failure rate, not simply the proportion of items that have failed.

The mean time to failure (MTTF) for a non-repairable item is

For a repairable item, mean time between failures (MTBF) is commonly used, but its interpretation depends on how operating time, repair time, and recurring failures are defined. MTBF is not automatically the life of every item, nor does it guarantee failure-free operation for that duration.

Exponential Reliability Model

If failures occur at a constant rate λ and successive conditions are appropriate, the exponential model gives

This model is suitable for a constant-hazard period, often associated with random failures, but not generally for early-life defects or wear-out.

Weibull Reliability Model

The Weibull model is flexible for life data:

where β > 0 is the shape parameter and η > 0 is the characteristic life, with the same time units as t. Its hazard is

If β < 1, hazard decreases with time, suggesting infant mortality. If β = 1, the model becomes exponential with constant hazard. If β > 1, hazard increases, suggesting wear-out. The Weibull model is not proof of a physical failure mechanism; its adequacy must be checked using data and engineering knowledge.

For a two-parameter Weibull distribution, the mean life is

where Γ is the gamma function.

Series and Parallel Systems

For statistically independent components in series, the system works only if every component works:

For two independent parallel components, the system works if at least one works:

These equations assume clearly defined system logic and statistical independence. Common-cause failures, shared power supplies, and environmental shocks can make the formulas optimistic.

Maintainability and Availability

Maintainability concerns the probability that a failed item is restored within a specified time. Under an exponential repair-time model with repair rate μᵣ,

where μᵣ has units of inverse time. A simple steady-state availability approximation is

where MTTR is mean time to repair, including relevant diagnosis and restoration time. This expression assumes a stable repairable system and clearly defined operating and repair periods.

Applied Case: Reliability and SPC in a CNC Turning Cell

Background

This illustrative case concerns a CNC turning cell producing precision shafts. The quality target is a diameter of 25.00 ± 0.05 mm. Historical records show occasional oversize parts and unplanned spindle-motor failures.

Problem

The production team proposes increasing preventive maintenance frequency, but maintenance consumes capacity. Before changing the schedule, the team must determine whether failures are increasing with age and whether dimensional variation is stable.

Analysis

Operators collect five consecutive diameter measurements every hour and plot subgroup means and ranges. The chart shows a sustained upward run in the means after approximately 1,200 spindle operating hours, while ranges remain broadly stable. This pattern indicates a centring change rather than a sudden increase in short-term measurement variation. Investigation finds progressive thermal drift associated with spindle-bearing wear.

A life-data analysis of 18 motors fits a Weibull model with estimated β = 2.4. Since β > 1, the failure hazard is increasing, consistent with wear-out. The characteristic life estimate is η = 8,000 operating hours. Reliability at 6,000 hours is

Thus, the estimated probability of surviving to 6,000 hours is about 61.1%, subject to the quality of the fitted model and limited sample size.

The engineering team compares replacement policies using expected production loss, spare-part cost, and maintenance labour. It schedules inspection at 5,000 hours and planned replacement near the age at which the economic risk of failure exceeds the cost of replacement. It also recalibrates the dimensional monitoring procedure and establishes a reaction plan for an out-of-control signal.

Outcome and Learning Points

The decision is not based on MTBF alone. SPC identifies a process signal before many parts become nonconforming, while Weibull analysis indicates an increasing failure hazard. The combined approach supports condition-based or age-informed maintenance rather than an arbitrary calendar interval.

The case demonstrates five principles:

1. Quality and reliability data should be analysed together when equipment condition affects product characteristics.
2. A control limit is not a specification limit.
3. A fitted distribution is a model, not an observed fact.
4. Maintenance decisions should include cost, safety, downtime, and uncertainty.
5. Data collection must record operating conditions, censoring, repairs, and measurement reliability.

Limitations, Ethics, and Contemporary Practice

Statistical methods can mislead when samples are biased, measurements are unstable, observations are dependent, or data are selectively removed. Censored reliability data—items that have not failed when observation ends—must not be treated as ordinary complete failure times. Preventive-maintenance records may also suffer from survivorship bias because failed machines receive more attention than healthy ones.

Measurement-system analysis is therefore essential. Repeatability describes variation under repeated measurement by the same operator and instrument; reproducibility describes variation between operators or conditions. A calibration certificate does not by itself prove that a measurement system is adequate for the tolerance being controlled.

Ethically, engineers should report uncertainty, missing data, excluded observations, and model assumptions. A control chart should never be manipulated to conceal an adverse signal. Automated and AI-based predictive-maintenance systems require traceable data, cybersecurity, access control, human review, and monitoring for model drift. Predictions should assist—not silently replace—competent engineering judgement, especially where failure creates safety or environmental risk.

Best practice includes defining the decision before collecting data, using a representative sampling plan, validating the measurement system, checking assumptions with plots and diagnostics, separating exploration from confirmation, documenting changes, and linking statistical signals to a physically plausible investigation.

Chapter Summary

Probability quantifies uncertain events through rules for unions, intersections, conditional probabilities, and Bayes’ theorem. Random variables and distributions such as binomial, Poisson, normal, and exponential models provide structured descriptions of manufacturing outcomes.

Sampling statistics estimate population characteristics, while confidence intervals express uncertainty. Hypothesis tests distinguish evidence against a specified null hypothesis, but practical importance and decision costs must also be considered. Correlation measures linear association; regression predicts conditional means but does not by itself establish causation.

Design of experiments uses randomization, replication, blocking, and factorial structures to estimate effects and interactions efficiently. SPC separates common-cause from special-cause variation, and capability indices evaluate stable processes against specifications. Reliability analysis uses survival, hazard, exponential, and Weibull models to understand failure behaviour. Series and parallel reliability equations describe system structures, while maintainability and availability connect failure behaviour to restoration performance.

In machine technology, the strongest decisions combine statistical evidence, engineering knowledge, economic analysis, and ethical reporting of uncertainty.

Key Terms

Probability: Numerical measure of the likelihood of an event.

Conditional probability: Probability of an event given that another event has occurred.

Random variable: Numerical representation of an uncertain outcome.

Expected value: Probability-weighted long-run average of a random variable.

Variance: Mean squared deviation from the mean.

Standard deviation: Square root of variance, expressed in the original units.

Population: Complete set of units or observations under study.

Sample: Subset selected from a population.

Confidence interval: Interval produced by a procedure designed to contain a parameter at a stated long-run confidence level.

Null hypothesis: Provisional statistical statement tested against data.

p-value: Probability, under the null hypothesis, of a result at least as extreme as observed.

Correlation: Measure of linear association between two variables.

Regression: Model relating a response to one or more explanatory variables.

Interaction: Dependence of one factor’s effect on another factor’s level.

Statistical process control: Use of data over time to monitor process behaviour.

Specification limit: Customer or design boundary for acceptable output.

Reliability: Probability of performing a required function for a stated time and condition.

Hazard rate: Conditional instantaneous rate of failure.

Maintainability: Probability of restoring an item within a stated time.

Availability: Proportion or probability of being operational when required.

Review Questions

Short Answer Questions

1. Distinguish between a population and a sample.
2. State the difference between a control limit and a specification limit.
3. What does a Weibull shape parameter β greater than 1 generally indicate?
4. Define a Type I error.
5. Why does correlation not prove causation?

Long Answer Questions

1. Explain Bayes’ theorem and describe its use in machine fault diagnosis.
2. Compare binomial, Poisson, normal, and exponential distributions, including one appropriate application for each.
3. Explain the assumptions and interpretation of a one-sample t test.
4. Describe the principles of randomization, replication, and blocking in design of experiments.
5. Explain how SPC and reliability analysis can be integrated in maintenance decision-making.

Analytical / Application Questions

1. A production process has a target diameter of 40.00 mm, with estimated μ = 40.03 mm and σ = 0.02 mm. Specification limits are 40.00 and 40.08 mm. Calculate Cₚ and Cₚₖ, and interpret the difference.
2. Two independent components have reliabilities 0.96 and 0.98 at 1,000 operating hours. Calculate the reliability of a series system and of a parallel system, and explain which architecture is more reliable.
3. A maintenance database contains failure times, several right-censored observations, repair durations, and machine operating loads. Propose a defensible analysis plan, identifying at least three assumptions or data-quality risks that must be checked before fitting a reliability model.

Multiple Choice Questions with Answers

Question 1

A machine inspection system has P(D)=0.02, P(+|D)=0.95, and P(+|not D)=0.03. What is approximately P(D|+)?

A. 0.020

B. 0.039

C. 0.393

D. 0.950

Correct Answer: C. 0.393

Question 2

If X follows a Poisson distribution with mean 1.5, what is P(X=2) approximately?

A. 0.112

B. 0.251

C. 0.500

D. 0.750

Correct Answer: B. 0.251

Question 3

For a sample of 16 observations with x̄=25.06 mm, s=0.08 mm, and μ₀=25.00 mm, what is the one-sample t statistic?

A. 0.75

B. 1.50

C. 2.13

D. 3.00

Correct Answer: D. 3.00

Question 4

Which statement about correlation is correct?

A. It proves that one variable causes the other.

B. It measures linear association and may be affected by outliers.

C. It can take values only from 0 to 1.

D. It is unaffected by the range of observed data.

Correct Answer: B. It measures linear association and may be affected by outliers.

Question 5

Which DOE principle reduces bias caused by time-related changes in an experiment?

A. Randomization

B. Extrapolation

C. Aggregation

D. Censoring

Correct Answer: A. Randomization

Question 6

A process has USL=10, LSL=8, and σ=0.25. What is Cₚ?

A. 0.67

B. 1.00

C. 1.33

D. 2.67

Correct Answer: C. 1.33

Question 7

For an exponential reliability model, the hazard rate is assumed to be:

A. Constant

B. Always increasing

C. Always decreasing

D. Equal to zero after the first failure

Correct Answer: A. Constant

Question 8

In a Weibull model, β>1 generally indicates:

A. Decreasing failure hazard

B. Constant failure hazard

C. Increasing failure hazard

D. No possible failures

Correct Answer: C. Increasing failure hazard

Question 9

Two independent components have reliabilities 0.9 and 0.8 in series. What is system reliability?

A. 0.10

B. 0.72

C. 0.85

D. 0.98

Correct Answer: B. 0.72

Question 10

Which statement about statistical significance is most accurate?

A. It always means the effect is economically important.

B. It is unrelated to sample size.

C. It indicates evidence against a null hypothesis under specified assumptions.

D. It proves the fitted model is physically correct.

Correct Answer: C. It indicates evidence against a null hypothesis under specified assumptions.