Probability is the mathematics of uncertainty; statistics uses data to make decisions despite it. Today this subject sits under A/B tests, alert thresholds, capacity planning, reliability targets and every machine-learning model. This article revisits the five units of the JNTUK course.
Unit 1: Probability
Basic definitions
| Term | Meaning |
|---|---|
| Experiment | A process with uncertain outcome (rolling a die) |
| Sample space S | The set of all outcomes |
| Event | A subset of the sample space |
| Probability P(A) | A number from 0 to 1; P(S) = 1 |
If all outcomes are equally likely, P(A) = favourable outcomes / total outcomes.
Rules
| Rule | Formula |
|---|---|
| Complement | P(not A) = 1 − P(A) |
| Addition | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| Conditional | P(A | B) = P(A ∩ B) / P(B) |
| Multiplication | P(A ∩ B) = P(A) P(B | A) |
| Independence | A and B are independent if P(A ∩ B) = P(A) P(B) |
Mutually exclusive events cannot happen together; that is different from independent.
Total probability and Bayes’ theorem
If events B₁, …, Bₙ partition the sample space, the total probability of A is Σ P(A | Bᵢ) P(Bᵢ). Bayes’ theorem reverses a conditional probability:
P(B | A) = P(A | B) P(B) / P(A)
Example: a disease affects 1% of people. A test detects it 99% of the time but also gives a false positive for 5% of healthy people. If you test positive, the chance you actually have it is 0.99 × 0.01 / (0.99 × 0.01 + 0.05 × 0.99) ≈ 17%, far lower than most people guess, because healthy people are so much more numerous.
Today: Bayes’ theorem is behind spam filters and diagnostic tools. It is also why an alert with a small false-positive rate can still be wrong most of the time when the thing it detects is rare.
Unit 2: Random variables and expectation
A random variable X assigns a number to each outcome. It is discrete (countable values, described by a probability mass function) or continuous (described by a probability density function f(x), where probabilities are areas under the curve). The cumulative distribution function F(x) = P(X ≤ x) works for both.
Expectation and variance
| Quantity | Discrete formula | Meaning |
|---|---|---|
| Mean E(X) | Σ x P(x) | The long-run average |
| Variance Var(X) | E(X²) − (E(X))² | Spread around the mean |
| Standard deviation | √Var(X) | Spread, in the units of X |
Properties: E(aX + b) = a E(X) + b; Var(aX + b) = a² Var(X); for independent variables the variances add. The moment generating function produces all moments of a distribution. Chebyshev’s inequality says at least 1 − 1/k² of any distribution lies within k standard deviations of the mean.
Today: an SLO error budget is an expected value; “p99 latency” is a point on the cumulative distribution. A mean can hide a long tail, which is why both mean and spread are reported.
Unit 3: Probability distributions
Discrete distributions
| Distribution | Models | Mean | Variance |
|---|---|---|---|
| Bernoulli | One yes/no trial with probability p | p | p(1 − p) |
| Binomial B(n, p) | Number of successes in n independent trials | np | np(1 − p) |
| Poisson(λ) | Number of events in a fixed interval, rare and independent | λ | λ |
| Geometric | Trials until the first success | 1/p | (1 − p)/p² |
The Poisson distribution approximates the binomial when n is large and p is small.
Continuous distributions
| Distribution | Models | Notes |
|---|---|---|
| Uniform | Every value in [a, b] equally likely | Mean (a + b)/2 |
| Exponential(λ) | Waiting time between Poisson events | Mean 1/λ; “memoryless” |
| Normal N(μ, σ²) | Bell curve; sums of many small effects | Symmetric; mean μ, variance σ² |
For the normal distribution, about 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ. Any normal variable is converted to the standard normal with Z = (X − μ) / σ and read from a Z-table. The central limit theorem: the average of many independent samples is approximately normal, whatever the original distribution.
Today: requests per second to a service are roughly Poisson, and the time between them is exponential, which is the basis of queueing models. Errors per thousand requests are binomial. The 3σ rule underlies simple anomaly-detection thresholds on metrics.
Unit 4: Sampling and estimation
Population and sample
A population is everything of interest; a sample is the part you observe. Parameters (μ, σ) describe populations; statistics (x̄, s) describe samples. The sample mean has its own distribution, with mean μ and standard error σ/√n, so larger samples give tighter estimates.
Estimation
| Idea | Meaning |
|---|---|
| Point estimate | A single value from the sample (x̄ for μ) |
| Unbiased estimator | Its average over many samples equals the true parameter (sample variance divides by n − 1) |
| Confidence interval | A range built to contain the true parameter with a stated probability |
| Maximum likelihood | Choose the parameter that makes the observed data most probable |
A 95% confidence interval for a mean (large sample) is x̄ ± 1.96 σ/√n. It does not mean “95% probability that the true mean is inside this particular interval”; it means the method captures the true value in 95% of repeated samples. For small samples with unknown σ, the t-distribution is used. Intervals for proportions work similarly.
Today: a load test run once gives a point estimate; running it ten times gives an interval. Survey and benchmark numbers without a sample size or interval should be treated with caution.
Unit 5: Hypothesis testing, correlation and regression
Hypothesis testing
You test a claim by asking whether the data would be surprising if the claim about “no effect” were true.
| Step | Detail |
|---|---|
| 1. Hypotheses | Null H₀ (no effect) versus alternative H₁ |
| 2. Significance level α | The acceptable false-alarm rate, usually 0.05 |
| 3. Test statistic | Z, t, χ² or F, from the data |
| 4. Decision | Compare with the critical value, or use the p-value: reject H₀ if p < α |
| Error | Meaning |
|---|---|
| Type I | Reject a true H₀ (false positive), probability α |
| Type II | Fail to reject a false H₀ (false negative), probability β |
Common tests: the Z-test and t-test (compare means), the chi-square test (categorical data, goodness of fit, independence) and the F-test (compare variances; ANOVA compares several means). A p-value is the probability of seeing data at least this extreme if H₀ is true; it is not the probability that H₀ is true.
Correlation and regression
Correlation r measures the strength of a linear relationship, from −1 to +1. Regression fits a line y = a + bx by least squares, with slope b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)². The coefficient of determination r² is the share of variation explained by the line. Correlation does not imply causation: two things can move together because of a third factor.
Queueing (as covered in the syllabus)
An M/M/1 queue has Poisson arrivals at rate λ and exponential service at rate μ. Utilization ρ = λ/μ; the average number in the system is ρ / (1 − ρ), which grows sharply as ρ approaches 1.
Today: A/B tests are hypothesis tests on conversion rates; flaky-test detection and regression checks on latency use the same logic. Linear regression is the simplest machine-learning model. The queueing result explains why a server at 90% utilization has far longer waits than one at 60%, and why autoscalers keep headroom.
One-minute recap
| Unit | Key idea | You meet it in |
|---|---|---|
| 1. Probability | Addition, conditional, Bayes’ theorem | Spam filters, alert false positives |
| 2. Random variables | Mean, variance, CDF | Latency percentiles, error budgets |
| 3. Distributions | Binomial, Poisson, exponential, normal; central limit theorem | Request arrivals, anomaly thresholds |
| 4. Sampling and estimation | Standard error, confidence intervals | Load tests, benchmarks, surveys |
| 5. Testing and regression | H₀, p-value, Type I and II errors, least squares | A/B tests, trend lines, capacity planning |
Next in Engineering Revisited: Computer Graphics.
Written by Claude, Anthropic’s AI assistant, as part of Engineering Revisited, a quick refresher of the B.Tech subjects one at a time.