14D

Venn diagrams and the addition theorem

Mutually exclusive events, the addition theorem P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and using Venn diagrams for "or" questions.

The addition theorem

Addition theorem

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

This is the counting rule divided by $|S|$. Events that cannot happen together are mutually exclusive: $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

Example 1●

A card is drawn. Find $P(\text{heart or king})$.

$P(\text{heart}) = \dfrac{13}{52}$, $P(\text{king}) = \dfrac{4}{52}$, $P(\text{king of hearts}) = \dfrac{1}{52}$.

$P = \dfrac{13 + 4 - 1}{52} = \dfrac{16}{52} = \dfrac{4}{13}$

Example 2●●

$P(A \cup B) = 0.8$, $P(A) = 0.5$ and $P(B) = 0.6$. Find $P(A \cap B)$ and $P(\text{neither})$.

$0.8 = 0.5 + 0.6 - P(A \cap B)$, so $P(A \cap B) = 0.3$.

$P(\text{neither}) = 1 - 0.8 = 0.2$

Flashcards

Addition theorem

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Mutually exclusive

$P(A \cap B) = 0$

$P( ext{neither})$

$1 - P(A \cup B)$

Practice

Find $P(\text{heart or king})$ for one card.
$P(A) = 0.5$, $P(B) = 0.3$, $P(A \cap B) = 0.1$. Find $P(A \cup B)$.
$A$ and $B$ are mutually exclusive when:
A die is rolled. Find $P(\text{even or greater than } 4)$.
$A$ and $B$ are mutually exclusive with $P(A) = 0.4$ and $P(B) = 0.35$. Find $P(A \cup B)$.
$P(A \cup B) = 0.8$, $P(A) = 0.5$, $P(B) = 0.6$. Find $P(A \cap B)$.
For the same events, find $P(\text{neither})$.
Which pair is mutually exclusive when one card is drawn?
A number is chosen from $1$ to $20$. Find $P(\text{multiple of 4 or multiple of 5})$.