Rules of Probability
Probability rules are fundamental principles that govern how probabilities are calculated and combined. These rules allow us to analyze complex events and make predictions in uncertain scenarios.
1. Addition Rule
The addition rule is used to calculate the probability of the union of two events (i.e., either event $A$ or event $B$, or both occurring).
Formula:
For two events $A$ and $B$:
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]Special Cases:
-
Mutually Exclusive Events: If $A$ and $B$ cannot occur simultaneously ($P(A \cap B) = 0$):
\[P(A \cup B) = P(A) + P(B)\]
Example:
A card is drawn from a deck. What is the probability of drawing a King or a Heart?
- $P(\text{King}) = \frac{4}{52}$
- $P(\text{Heart}) = \frac{13}{52}$
- $P(\text{King} \cap \text{Heart}) = \frac{1}{52}$ (King of Hearts)
2. Multiplication Rule
The multiplication rule is used to calculate the probability of the intersection of two events (i.e., both event $A$ and event $B$ occurring).
Formula:
\[P(A \cap B) = P(A) \cdot P(B \mid A)\]Where $P(B \mid A)$ is the conditional probability of $B$ given $A$.
Special Cases:
-
Independent Events: If $A$ and $B$ are independent ( $P(B \mid A) = P(B)$ ):
\[P(A \cap B) = P(A) \cdot P(B)\]
Example:
Tossing two coins. What is the probability of getting heads on both coins?
- $P(\text{Heads on 1st coin}) = \frac{1}{2}$
- $P(\text{Heads on 2nd coin}) = \frac{1}{2}$
3. Complement Rule
The complement rule calculates the probability of an event not occurring.
Formula:
\[P(\text{Not A}) = 1 - P(A)\]Example:
What is the probability of not rolling a 6 on a fair die?
\[P(\text{Not 6}) = 1 - P(6) = 1 - \frac{1}{6} = \frac{5}{6}\]4. Conditional Probability
Conditional probability calculates the probability of an event $B$ given that another event $A$ has occurred.
Formula:
\[P(B \mid A) = \frac{P(A \cap B)}{P(A)}\]Provided $P(A) > 0 $.
Example:
A card is drawn from a deck. What is the probability that it is a King given it is a Heart?
- $P(\text{King} \cap \text{Heart}) = \frac{1}{52}$
- $P(\text{Heart}) = \frac{13}{52}$
Summary of Rules
| Rule | Formula | Key Use Case |
|---|---|---|
| Addition Rule | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | For “or” probabilities |
| Multiplication Rule | $P(A \cap B) = P(A) \cdot P(B \mid A)$ | For “and” probabilities |
| Complement Rule | $P(\text{Not A}) = 1 - P(A)$ | For probabilities of the complement |
| Conditional Probability | $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$ | For probabilities given specific conditions |
Applications in Real Life
-
Addition Rule:
Calculating the probability of a card being a King or a Queen. -
Multiplication Rule:
Determining the likelihood of two events happening in sequence, such as drawing two aces in a row. -
Complement Rule:
Estimating the probability of not getting a defective product in a batch. -
Conditional Probability:
Assessing the likelihood of disease given a positive test result.
Conclusion
The rules of probability are essential tools for analyzing uncertain events and making data-driven predictions. Mastering these rules allows you to handle more complex probabilistic scenarios with confidence.
Next Steps: Discrete and Continuous Distributions