ENGR 200 — First Midterm Study Guide

What this is: A concise guide to the Conceptual and Calculation topics you need for the midterm.

Conceptual Part — Chapters 2 & 3

Chapter 2: Probability Foundations

Core Rules
  • Bounds: 0 ≤ P(A) ≤ 1
  • Total probability: P(S) = 1
  • Complement: P(Ac) = 1 − P(A)
  • Addition (2 sets): P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independence vs. Mutual Exclusivity
  • Independent: P(A ∩ B) = P(A)P(B). Knowing one doesn’t change the other.
  • Mutually exclusive (disjoint): P(A ∩ B) = 0. Both can’t happen at once.
  • Important: If P(A) > 0 and P(B) > 0, disjoint ≠ independent.
Counting Models
  • Permutation (order matters): P(n,r) = n! / (n − r)!
  • Combination (order doesn’t): C(n,r) = n! / [r!(n − r)!]
  • Heuristic: roles/arrangements → permutation; groups/sets → combination.
Total Probability & Bayes
  • Total probability: P(A) = Σ P(A|Bi)P(Bi)
  • Bayes’ theorem: P(Bj|A) = [P(A|Bj)P(Bj)] / Σ P(A|Bi)P(Bi)
  • Always compute the denominator (P(A)) as the sum of matches.

Chapter 3: Identify the Right Model

Binomial
  • Fixed trials n, constant success prob p.
  • Counts successes in n.
  • Example: number of sixes in 50 rolls.
Poisson
  • Counts events in time/space with rate λ.
  • Unbounded counts: 0,1,2,…
  • Example: calls in the next hour.
Geometric
  • Trials to first success; constant p.
  • Support: 1,2,3,… (includes the success trial).
  • Example: rolls until first “1”.
Uniform (continuous)
  • All values equally likely on [a,b].
  • Density: f(x) = 1 / (b − a).
  • Example: arrival time uniformly in a 10-min window.
Model ID strategy: Ask 3 questions — (1) Are outcomes discrete counts or a continuous value? (2) Is there a fixed number of trials n with success probability p (Binomial) or a rate in time/space λ (Poisson)? (3) Are you waiting for the first success (Geometric)?

Calculation Part — A/B/C

Part A: Bayes’ Theorem

Workflow
  1. Set up rows for categories Bi with prior P(Bi) and likelihood P(A|Bi).
  2. Compute matches: mi = P(A|Bi)P(Bi).
  3. Sum matches: P(A) = Σ mi.
  4. Posterior: P(Bj|A) = mj / P(A).
Common checks
  • Posteriors across all cases sum to 1.
  • Units: percents vs decimals (be consistent).
  • Every case must have a prior and a likelihood.

Part B: Permutations & Combinations

Formulas
  • Permutations: P(n,r) = n! / (n − r)!
  • Combinations: C(n,r) = n! / [r!(n − r)!]
When to use
  • Roles/arrangements/schedules → permutations.
  • Groups/committees/selections → combinations.
  • Replace vs. without replacement: affects counts.

Part C: Discrete Models

Poisson Distribution
  • Probability mass function: P(X = k) = (e−λ λk) / k!
  • Mean parameter: λ = (rate) × (interval)
  • Shortcut: At least one event = 1 − P(0) = 1 − e−λ
Binomial Distribution
  • Probability mass function: P(X = k) = C(n,k) pk (1 − p)n−k
  • Use for: exact successes, cumulative (≤ or ≥), or complements.
Geometric Distribution
  • Probability mass function: P(X = k) = (1 − p)k−1 p
  • Represents the trial number of the first success.
  • Support: k = 1, 2, 3, …
Parameter hygiene: Define n, p, λ before you compute. Convert units (minutes → hours) when building λ. Use complements to simplify cumulative probabilities.

Smart Exam Tips

Quick Checklist

Good luck! Master the patterns above and you’ll be ready for both the conceptual and calculation parts.