Probability Fundamentals

FE Exam Weight: Probability and Statistics accounts for 6-9 questions (~7% of the exam). These questions are often straightforward if you know the formulas.

Basic Probability Rules

Sample Space (S): The set of all possible outcomes.

Event (A): A subset of the sample space.

$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Axioms of Probability

1. 0 ≤ P(A) ≤ 1 for any event A
2. P(S) = 1 (something must happen)
3. P(A') = 1 - P(A) where A' is the complement of A

Addition Rule

• General: P(A or B) = P(A) + P(B) - P(A and B)
• Mutually exclusive events (A ∩ B = ∅): P(A or B) = P(A) + P(B)

Multiplication Rule

• General: P(A and B) = P(A) · P(B|A)
• Independent events: P(A and B) = P(A) · P(B)

Conditional Probability

The probability of A given that B has occurred: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Example: A box contains 3 red and 7 blue balls. Two balls are drawn without replacement. What is the probability the second ball is red given the first was blue?

P(R₂|B₁) = 3/9 = 1/3 (3 red remain out of 9 total)

Bayes' Theorem

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Extended form (with multiple hypotheses): $P(A_i|B) = \frac{P(B|A_i) \cdot P(A_i)}{\sum_j P(B|A_j) \cdot P(A_j)}$

Engineering Application: A manufacturing process produces 2% defective parts. A test detects defectives with 95% accuracy and falsely identifies good parts as defective 3% of the time. If a part tests positive, what is the probability it is actually defective?

P(D) = 0.02, P(+|D) = 0.95, P(+|D') = 0.03

P(D|+) = (0.95 × 0.02) / (0.95 × 0.02 + 0.03 × 0.98) = 0.019 / 0.0484 = 0.393

Even with a positive test, the part is only ~39% likely to be defective because defectives are rare.

Counting Methods

Permutations (Order Matters)

$P(n, r) = \frac{n!}{(n-r)!}$

Example: How many ways can 3 people be selected for President, VP, and Treasurer from 10 candidates? P(10, 3) = 10!/(10-3)! = 10 × 9 × 8 = 720

Combinations (Order Does Not Matter)

$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

Example: How many ways can a committee of 3 be chosen from 10 people? C(10, 3) = 10!/(3!·7!) = 120