Probability and Statistics
Probability and statistics let engineers quantify uncertainty and make data-driven decisions about quality, reliability, and risk. The FE tests basic probability rules, descriptive statistics, the common distributions, and elementary estimation — mostly compact calculations where choosing the correct formula is everything.
Probability fundamentals
For equally likely outcomes, probability is favorable outcomes divided by total outcomes. A single die has P(roll > 4) = 2/6 = 1/3. Two fair dice give 36 outcomes; the sum of 7 occurs 6 ways (P = 6/36 = 1/6), the sum of 8 occurs 5 ways (P = 5/36). A king drawn from 52 cards is 4/52 = 1/13. The complement rule P(not A) = 1 − P(A) often shortcuts 'at least one' problems. Keeping the sample space complete and correctly counted prevents most errors.
Combining events: AND and OR
For independent events, joint probability multiplies: P(A and B) = P(A)·P(B). For mutually exclusive events, probabilities add: P(A or B) = P(A) + P(B). When events can overlap, use inclusion–exclusion: P(A or B) = P(A) + P(B) − P(A and B). Conditional probability is P(A|B) = P(A and B)/P(B), and Bayes' theorem inverts conditionals. The decision that trips students is whether events are independent or mutually exclusive — these are different properties, and applying the wrong combining rule gives a wrong answer that often still 'looks reasonable.'
Descriptive statistics
The mean x̄ = Σxᵢ/n locates the center; the median is the middle value (robust to outliers); the mode is the most frequent. Spread is measured by variance and its square root, the standard deviation. Crucially, the SAMPLE standard deviation divides the summed squared deviations by (n − 1), not n, to remove bias: for 2, 4, 6, 8, 10 the mean is 6, the squared deviations sum to 40, and s = √(40/4) = √10 ≈ 3.16. Using n instead of (n − 1) for sample data is the single most common statistics mistake on the FE.
Discrete distributions
The binomial distribution models the number of successes in n independent trials each with success probability p: P(k) = C(n,k)·pᵏ·(1−p)^(n−k), with mean μ = n·p and variance σ² = n·p·(1−p). Three coin flips giving exactly two heads is C(3,2)·(0.5)³ = 3/8. With n = 20 and p = 0.3 the mean is 6; with n = 100 and p = 0.5 the variance is 25. The Poisson distribution models counts of rare events over an interval, with mean and variance both equal to λ. Matching the experiment to the right discrete model is the skill tested.
The normal distribution
The normal (Gaussian) distribution is the bell curve defined by its mean μ and standard deviation σ. The empirical rule places about 68% of data within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Any value is standardized to a z-score z = (x − μ)/σ, which is then looked up in a standard-normal table to find tail probabilities. By the central limit theorem, the mean of a large sample is approximately normal regardless of the parent distribution — the reason the normal appears so widely in quality control and measurement error.
Estimation and hypothesis testing
A confidence interval brackets an unknown population parameter with a stated confidence, e.g. x̄ ± z·(σ/√n) for a mean; larger samples shrink the interval because the standard error falls as 1/√n. Hypothesis testing weighs a null hypothesis against an alternative, rejecting the null when a test statistic falls in the critical region. A Type I error rejects a true null (probability α, the significance level); a Type II error fails to reject a false null. These ideas connect sampled data back to decisions about the underlying process.
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