📘 Chapter 2: Probability Concepts for Engineers

The union A ∪ B means A occurs, or B occurs, or both occur.

The intersection A ∩ B means both A and B occur together.

Always 1, because some outcome must happen in every trial.

It is all outcomes in the sample space that are not in A, written as A′.

If A and B are independent, then P(A ∩ B) = P(A) × P(B).

The probability of A given that B has occurred: P(A|B) = P(A ∩ B) / P(B).

It allows us to update the probability of a cause (event) when new evidence is observed.

Permutation cares about order of selection; Combination does not.

1 − P(no heads) = 1 − (1/2 × 1/2) = 3/4.

They help model uncertainty in real systems, reliability analysis, and risk management.

• Independent: Knowing one does not change the other. P(A∩B) = P(A)P(B) (equivalently, P(A|B)=P(A) and P(B|A)=P(B)). Independent events often do overlap.
• Mutually exclusive (disjoint): Cannot occur together. P(A∩B) = 0. If P(A)>0 and P(B)>0, then they are automatically not independent because P(A)P(B)>0 ≠ 0.

Examples: One coin toss—A: “Heads”, B: “Tails” ⇒ mutually exclusive (and dependent). Two coins—A: “1st is Heads”, B: “2nd is Heads” ⇒ independent.

What it does: Bayes converts likelihoods into an updated belief (posterior): P(A|B) = [P(B|A)·P(A)] / P(B). It “turns around” conditional probability to answer “How likely is the cause given what I saw?”

Why important: It’s the engine behind diagnostic testing, spam filtering, reliability and fault attribution, sensor fusion, and many ML workflows where you update prior knowledge with new evidence.

Quick example (classroom pepper): Let A = “Foreign”, B = “Likes pepper”. Assume P(A)=0.20, P(B|A)=0.80, P(B|A′)=0.30. Then P(B)=0.80·0.20 + 0.30·0.80 = 0.40. Thus P(A|B) = (0.80·0.20)/0.40 = 0.40. Interpretation: among pepper-lovers, 40% are Foreign, 60% USA.