Source: Principles of Statistics I, Texas A&M University
Tags: general probability, marginal probability, joint probability, complement, union, intersection, addition rule, Venn diagram, contingency table, two-way table, sample space
General probability covers how to calculate the likelihood of events from tabulated data or given information. The core toolkit is the addition rule, complements, and intersections, all of which you can read straight off a two-way table or Venn diagram once the setup is right.
Sample space
The set of all possible outcomes for a random experiment. Every probability you calculate lives inside this space.
Event
A subset of the sample space. Labelled with capital letters (A, B, etc.) and defined in words so you know exactly which outcomes belong to it.
Complement (A')
Everything in the sample space that is not in A. Calculated as P(A') = 1 − P(A). Sometimes written as A^c.
Union (A ∪ B)
The event that A or B (or both) occur. "At least one of the two."
Intersection (A ∩ B)
The event that both A and B occur simultaneously. The overlap region in a Venn diagram.
Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You subtract the intersection to avoid counting it twice.
Marginal probability
The row total or column total in a two-way table, divided by the grand total. It gives the overall probability of a single event without conditioning on anything else.
Joint probability
The probability of two events occurring together, i.e. a single cell in the two-way table divided by the grand total.
A two-way (contingency) table cross-classifies individuals by two categorical variables.
Each cell count divided by the grand total gives the joint probability of that row-column combination.
Row totals and column totals divided by the grand total give marginal probabilities.
Worked example (hypertension and smoking)
The table has 180 individuals classified by hypertension status (yes/no) and smoking level (non, moderate, heavy).
Non-Smokers | Moderate | Heavy | Row total | |
|---|---|---|---|---|
Hypertensive | 21 | 36 | 30 | 87 |
Not Hypertensive | 48 | 26 | 19 | 93 |
Column total | 69 | 62 | 49 | 180 |
P(Heavy smoker) = (30 + 19) / 180 = 49/180 ≈ 0.272
Add both cells in the Heavy column, then divide by the grand total.
P(Not hypertensive) = 1 − P(Hypertensive) = 1 − 87/180 = 93/180 ≈ 0.517
Use the complement rather than adding the row manually (though both work).
P(Heavy smoker AND not hypertensive) = 19/180 ≈ 0.106
A single cell divided by the grand total gives the joint probability directly.
When you are given P(A), P(B), and P(A ∪ B) but not the intersection, rearrange the addition rule.
Worked example (traffic signals)
A motorist passes two signals. P(A) = 0.4, P(B) = 0.5, P(A ∪ B) = 0.6.
P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.4 + 0.5 − 0.6 = 0.3
Stopping at both signals.
P(A ∩ B') = P(A) − P(A ∩ B) = 0.4 − 0.3 = 0.1
Stopping at the first only. Remove the overlap from A's total probability.
P(Exactly one signal) = P(A ∪ B) − P(A ∩ B) = 0.6 − 0.3 = 0.3
"Or but not both" means strip the intersection out of the union.
Draw two overlapping circles for A and B.
Fill in the intersection first (P(A ∩ B)).
The A-only region is P(A) − P(A ∩ B).
The B-only region is P(B) − P(A ∩ B).
The outside region is 1 minus the union.
Every probability question can then be read off the diagram by adding the right regions.
Complement rule: P(A') = 1 − P(A)
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Rearranged for intersection: P(A ∩ B) = P(A) + P(B) − P(A ∪ B)
"A only" (A and not B): P(A ∩ B') = P(A) − P(A ∩ B)
Exactly one of A or B (symmetric difference): P(A ∪ B) − P(A ∩ B)
⚠️ The addition rule is the single most tested formula in this section. Know it cold and be comfortable rearranging it for the intersection.
⚠️ "At least one" means union. "Exactly one" means union minus intersection. Mixing these up is one of the most common exam errors.
⚠️ When reading a two-way table, double-check whether a question asks for a marginal (row/column total) or a joint (single cell). The denominator is always the grand total for unconditional probabilities.
⚠️ The complement shortcut (1 − P) saves time and avoids arithmetic mistakes when "not" is involved.
Q: In a two-way table with 200 people, 60 are in category A and 45 are in category B, with 20 in both. What is P(A ∪ B)?
A: P(A ∪ B) = 60/200 + 45/200 − 20/200 = 85/200 = 0.425
Q: If P(A) = 0.35 and P(A ∪ B) = 0.70 and P(A ∩ B) = 0.15, what is P(B)?
A: Rearrange the addition rule: P(B) = P(A ∪ B) − P(A) + P(A ∩ B) = 0.70 − 0.35 + 0.15 = 0.50
Q: A student is asked for P(A ∩ B') from a Venn diagram. They subtract P(B) from P(A). Why is that wrong?
A: You must subtract P(A ∩ B), not P(B). P(A ∩ B') = P(A) − P(A ∩ B). Subtracting all of P(B) removes parts of B that were never inside A to begin with.
Q: Given P(X) = 0.6 and P(X') = 0.4, what does P(X) + P(X') equal and why?
A: It equals 1.0, because X and X' are complements and together they cover the entire sample space.
general probability, basic probability, addition rule, inclusion-exclusion, union, intersection, complement rule, marginal probability, joint probability, two-way table, contingency table, Venn diagram, sample space, mutually exclusive, "at least one", "exactly one", symmetric difference, STAT 301, Principles of Statistics I, Texas A&M