Set Functions, Sample Spaces, and Axioms of Probability | Free Exam P Lesson

A batch of 500 wafers has 80 with scratches, 65 with misalignment, and 30 with both. The fraction with at least one defect is not $80 + 65 = 145$ but $80 + 65 - 30 = 115$ . Forget the overlap and you overcount by 26%.

A sample space $S$ is the set of all possible outcomes of a random experiment. Every subset of $S$ is an event.

You need four set operations:

Union $A \cup B$ , outcomes in $A$ or $B$ or both
Intersection $A \cap B$ , outcomes in both $A$ and $B$
Complement $A^{c}$ , outcomes in $S$ not in $A$
Difference $A ∖ B = A \cap B^{c}$ , outcomes in $A$ but not $B$

For two events, the Venn diagram has four mutually exclusive regions:

$A \cap B$ , both occur
$A \cap B^{c}$ , only $A$

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Common mistakes

Forgetting to subtract the intersection. With $P (A) = 0.12$ and $P (B) = 0.09$ , writing $P (A \cup B) = 0.21$ double-counts the overlap. Trap: 0.21 instead of 0.17.
Sign error in three-event inclusion-exclusion. The triple intersection $P (A \cap B \cap C)$ must be added back. Omitting it gives $P (A \cup H \cup U) = 0.75$ . Trap: 0.75 or its complement 0.25.
Confusing "at least one" with "exactly one." $P (at least one) = P (A \cup B) = 0.17$ . $P (exactly one) = P (A \cup B) - P (A \cap B) = 0.13$ . Trap: giving 0.17 when asked for exactly one.

Bottom line

Inclusion-exclusion: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ ; for three events add back the triple intersection.
Complement rule: $P (A^{c}) = 1 - P (A)$ , fastest path to "at least one" or "none."
De Morgan's Laws: $(A \cup B)^{c} = A^{c} \cap B^{c}$ and $(A \cap B)^{c} = A^{c} \cup B^{c}$ ; complementing flips the operation.
Partition formula: if $B_{1}, \dots, B_{n}$ partition $S$ , then $P (A) = \sum P (A \cap B_{i})$ .

Exam shortcut

When you see two or three named events with given probabilities, draw the Venn diagram immediately, fill the innermost intersection first, subtract outward, and verify all regions sum to 1. "PIE" for inclusion-exclusion: Plus singles, Intersection pairs subtracted, Everything-together added back. "Break the bar, flip the sign" for De Morgan's: complementing across a union flips it to an intersection (and vice versa).

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