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FE Mechanical study materials
FE Mechanical study resources

FE Mechanical: Probability and Statistics

FE Mechanical · Work through your exam, one topic at a time.

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Probability and Statistics

Probability and Statistics

The FE Mechanical probability and statistics items cover descriptive statistics, probability rules, common distributions, sampling, confidence intervals, hypothesis testing, regression, and quality control. The exam favors applied calculations: given a mean and standard deviation, find a probability; given sample data, choose the correct interval or test.

Core rules and concepts

  • Addition rule: P(A or B) = P(A) + P(B) - P(A and B). If A and B are mutually exclusive, the intersection term is zero.
  • Independence: P(A and B) = P(A)P(B). Do not confuse independence with mutual exclusivity; mutually exclusive events with nonzero probabilities are dependent.
  • Conditional probability: P(A|B) = P(A and B)/P(B). Bayes' theorem follows from this definition.
  • Sample variance uses n - 1 in the denominator: s² = Σ(x - x̄)²/(n - 1). Population variance uses N.
  • Normal distribution: z = (x - μ)/σ. The standard normal table gives the area to the left of z. About 68% of values lie within one standard deviation, 95% within two, and 99.7% within three.
  • Binomial: P(X = k) = C(n,k) p^k (1-p)^(n-k). Poisson: P(X = k) = e^(-λ) λ^k / k!. Exponential mean = 1/λ.
  • Confidence interval for a mean with known σ: x̄ ± z(α/2) σ/sqrt(n). With unknown σ and small n, use t.
  • Hypothesis test: reject H0 when p-value ≤ α. Type I error = α; Type II error = β; power = 1 - β.

Worked example

Tensile strength is normally distributed with mean 50 ksi and standard deviation 4 ksi. Find the probability that a specimen exceeds 56 ksi. z = (56 - 50)/4 = 1.5. From the standard normal table, the area to the left of 1.5 is 0.9332, so the area to the right is 1 - 0.9332 = 0.0668, or 6.68%.

Traps that cost marks

  • Using n instead of n - 1 for sample variance.
  • Treating mutually exclusive events as independent.
  • Interpreting a p-value as the probability that H0 is true.
  • Using a one-tailed critical value for a two-tailed test.
  • Using z when σ is unknown and the sample is small; use t.
  • Claiming causation from a high correlation coefficient.