Statistics / Unit 6: Probability
6.2 Complements and the Addition Rule
Notes
- Complement rule: $P(A^c) = 1 - P(A)$, where $A^c$ means "not $A$".
- Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
- Events are mutually exclusive (disjoint) if they cannot happen together, so $P(A \text{ and } B) = 0$ and $P(A \text{ or } B) = P(A) + P(B)$.
- Example: In a deck of 52 cards, $\displaystyle P(\text{heart or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$, since the king of hearts is counted twice otherwise.
Practice
The probability of rain tomorrow is $0.35$. The probability of no rain is:
$1 - 0.35 = 0.65$.
One card is drawn from a standard deck of 52. What is the probability it is a heart or a king?
$\displaystyle \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$.
$P(A) = 0.5$, $P(B) = 0.3$ and $P(A \text{ and } B) = 0.1$. What is $P(A \text{ or } B)$?
$0.5 + 0.3 - 0.1 = 0.7$.
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