CELE Engineering Mathematics — Engineering Data Analysis (Probability and Statistics)Revision Notes

Formulas

Exam tips

• Always re-read the problem: 'population' → divide by n; 'sample' → divide by (n − 1).
• For small odd-numbered data sets, the median is the ((n+1)/2)th value — memorize this.
• When computing sample variance by hand, use the computational formula s² = [Σx² − (Σx)²/n] / (n−1) to reduce arithmetic errors.
• Board exams frequently give data sets of 5 to 7 numbers — practice these by hand.
• If asked for 'the most appropriate measure of central tendency' for skewed data, the answer is the median.

Key points

• The three measures of central tendency are the mean (arithmetic average), median (middle value when data are ordered), and mode (most frequently occurring value).
• For a data set 2, 4, 6, 8, 10: mean = 6, median = 6, mode = none (all values appear once).
• Population variance uses divisor n; sample variance uses divisor (n − 1) — the Bessel correction that accounts for estimating a population parameter from a sample.
• Standard deviation (σ or s) is the square root of variance and has the same units as the original data, making it more interpretable than variance.
• When data are grouped, use the class midpoint as the representative value for each class interval.
• Coefficient of variation CV = (σ/μ) × 100% allows comparison of dispersion across data sets with different units or magnitudes.
• The range = maximum − minimum is the simplest but least informative measure of spread.

Definitions

Common mistakes

• Using divisor n instead of (n − 1) when the problem states 'sample standard deviation' — this is a very frequent board exam trap.
• Forgetting to order the data before finding the median.
• Confusing 'no mode' with a mode of zero.
• Computing variance but reporting it as standard deviation without taking the square root.
• Not using class midpoints when computing the mean of grouped frequency data.

Formulas

Exam tips

• Key words: 'OR' → addition rule; 'AND' → multiplication rule; 'given that' → conditional probability.
• For reliability problems with independent components: series system P(works) = product of all; parallel system P(fails) = product of individual failure probabilities.
• When drawing without replacement, probabilities change with each draw — use conditional probability.
• Card problems: 52 cards total; 4 suits of 13 each; 4 of each face value — memorize the deck structure.
• For dice: two dice → 36 total outcomes; enumerate favorable outcomes systematically.

Key points

• Probability is a number between 0 (impossible) and 1 (certain); P(sample space) = 1.
• The classical definition: P(A) = (number of favorable outcomes) / (total number of equally likely outcomes).
• The complement rule is one of the most powerful tools: P(not A) = 1 − P(A).
• Two events are mutually exclusive if they cannot occur simultaneously: P(A ∩ B) = 0.
• Two events are independent if the occurrence of one does not affect the other: P(A ∩ B) = P(A) × P(B).
• Conditional probability P(A|B) = P(A ∩ B) / P(B) — the probability of A given that B has occurred.
• Mutually exclusive ≠ independent. If A and B are mutually exclusive and both have P > 0, they are NOT independent.
• For 'at least one' problems, use complement: P(at least one) = 1 − P(none).

Definitions

Common mistakes

• Treating mutually exclusive events as independent — these are completely different concepts.
• Forgetting to subtract P(A ∩ B) in the addition rule when events overlap (e.g., king AND heart).
• Using the multiplication rule P(A)P(B) for dependent events (e.g., drawing without replacement).
• Adding probabilities instead of multiplying for 'A AND B' with independent events.
• Not recognizing that 'at least one' problems are most efficiently solved using the complement.

Formulas

Exam tips

• Quick check: 'arrange/rank/order/schedule/assign positions' → Permutation; 'choose/select/committee/team/group' → Combination.
• C(n, r) = C(n, n−r): choosing 3 to include is the same as choosing (n−3) to exclude — use this to simplify large calculations.
• P(n, n) = n! — arranging ALL n objects in a line.
• For board exam speed: cancel factorials algebraically before multiplying — e.g., 9!/6! = 9×8×7 directly.
• Lotto 6/42 problems: C(42,6) = 5,245,786 — know how to compute this rapidly.

Key points

• The fundamental principle of counting: if event A can occur in m ways and event B in n ways, together they occur in m × n ways.
• Permutations count arrangements where ORDER MATTERS — think of ranking, positioning, assigning roles.
• Combinations count selections where ORDER DOES NOT MATTER — think of committees, teams, groups.
• Factorial: n! = n × (n−1) × (n−2) × … × 2 × 1; by definition, 0! = 1.
• Distinguishing permutation vs. combination is the single most important skill in this topic area.
• Permutations of identical objects: n! / (n₁! n₂! … nₖ!) accounts for repeated elements.
• Circular permutations: (n−1)! — one element is fixed to remove rotational equivalence.

Definitions

Common mistakes

• Using permutation formula when combination is required (and vice versa) — always ask 'does order matter?'
• Forgetting that 0! = 1, which is needed when r = n in permutation problems.
• Not dividing by repeated-element factorials when arranging objects that are not all distinct.
• Confusing circular permutation (n−1)! with linear permutation n!.
• In lotto-type problems, automatically assuming order does not matter — verify from context.

Formulas

Exam tips

• Binomial checklist: (1) fixed n, (2) two outcomes, (3) constant p, (4) independent. All four must be true.
• Normal distribution: ALWAYS sketch the bell curve and shade the region before computing z-scores.
• Symmetry of normal: P(Z < −a) = P(Z > a) = 1 − P(Z < a). This halves your table lookups.
• Poisson: if the problem gives an average rate (events per unit) and asks for exact count probability → Poisson.
• 68-95-99.7 rule: if asked about probability within 1, 2, or 3 standard deviations — use this rule directly without table.
• Binomial mean np is the expected value — also the answer to 'how many are expected to succeed?'

Key points

• A probability distribution describes all possible outcomes of a random variable and their associated probabilities.
• The Binomial distribution applies when: (1) fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) P(success) = p is constant, (4) trials are independent.
• The Normal (Gaussian) distribution is symmetric and bell-shaped, fully described by its mean μ and standard deviation σ.
• The Empirical Rule (68-95-99.7 Rule): approximately 68% of data fall within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
• The Standard Normal Distribution has μ = 0 and σ = 1; standardize any normal variable using z = (x − μ)/σ.
• The Poisson distribution models the number of rare, random events occurring in a fixed interval of time or space.
• For the binomial distribution: mean μ = np; variance σ² = np(1−p); std dev σ = √[np(1−p)].
• For the Poisson distribution: mean = λ; variance = λ (mean equals variance — a key identifying feature).

Definitions

Common mistakes

• Applying the binomial formula when trials are not independent (e.g., drawing without replacement from a small population — use hypergeometric instead).
• Forgetting to check all four conditions of the binomial distribution before applying its formula.
• Using the wrong tail of the z-table: P(X < a) vs P(X > a) vs P(a < X < b) — sketch the curve first.
• Confusing P(X = x) with P(X ≤ x) — binomial point probability vs. cumulative probability.
• Misidentifying a Poisson problem as binomial because the problem mentions 'probability' — look for rare events over an interval.
• Not converting negative z-scores correctly: P(Z < −1) = P(Z > 1) by symmetry = 1 − P(Z < 1).