Probability foundations

Learning objectives

• Apply complement, addition and multiplication rules of probability.
• Distinguish independent and conditional events with garment-line evidence.
• Build simple contingency tables in a spreadsheet.

Core idea

Probability lies in [0,1]. P(not A)=1−P(A). For unions P(A∪B)=P(A)+P(B)−P(A∩B). For independent events P(A∩B)=P(A)·P(B); conditional P(A|B)=P(A∩B)/P(B).

Spreadsheet lab

• Build a 2×2 table of seam vs measurement failures using =COUNTIFS.
• Row totals, column totals and grand total drive every probability.
• P(defect) = defects / total; P(repair | defect) = repairs∩defects / defects.
• Independence check: |P(A|B) − P(A)| near zero.

Garment-factory example

In 500 inspected polos, 40 have seam defects, 25 have measurement failures and 8 have both. P(seam∪measurement) = (40+25−8)/500 = 0.114; the overlap must be subtracted or the risk is overstated.

Method

1. Define event, population and time window.
2. Build a contingency table with COUNTIFS.
3. State whether events are independent and why.

Common mistakes

• Adding overlapping event probabilities without subtracting the intersection.
• Assuming independence when the same operator drives both events.