What is Intersection, union, complement? — Mathematics

Definition 9.5 High School Mathematics · Chapter 9 — Probability and Sampling

Let $A$ and $B$ be events.

$A \cap B$ (“ $A$ and $B$ ”) occurs when both occur;
$A \cup B$ (“ $A$ or $B$ ”) occurs when at least one occurs;
the complement $\bar A$ (“not $A$ ”) occurs exactly when $A$ does not;
$A$ and $B$ are incompatible when they cannot occur together: $A \cap B = \varnothing$ .

Two overlapping events: in (A) + (B), the shaded intersection A ∩ B is counted twice, which explains the subtraction in the addition rule. — Two overlapping events: in $\P(A) + \P(B)$ , the shaded intersection $A \cap B$ is counted twice, which explains the subtraction in the addition rule.

Examples

Example 9.7

Draw one card from a standard $52$ -card deck. Let $A$ : “the card is a heart” ( $13$ cards) and $B$ : “the card is a king” ( $4$ cards). Then $A \cap B$ is “the king of hearts” ( $1$ card), and

\P(A \cup B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}.

The complement rule is often the fastest route: the probability that the card is not a heart is $1 - \frac{13}{52} = \frac34$ .

Example 9.9

An urn contains $3$ red balls and $2$ blue balls. Draw one ball, put it back, draw again. Each draw gives red (R) with probability $\frac35$ and blue (B) with probability $\frac25$ .

The tree of two draws with replacement. The four path probabilities sum to 1. — The tree of two draws with replacement. The four path probabilities sum to $1$ .

The probability of getting two balls of the same color is $\P(\text{RR}) + \P(\text{BB}) = \frac{9}{25} + \frac{4}{25} = \frac{13}{25}$ ; the probability of at least one blue is $1 - \P(\text{RR}) = 1 - \frac{9}{25} = \frac{16}{25}$ (complement rule).

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Examples

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