BackFundamental Probability Rules and Concepts
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Probability
Types of Probability
Probability is a measure of how likely an event is to occur. There are several ways to interpret and assign probabilities:
Theoretical Probability: When probability is based on a model with equally likely outcomes, it is called theoretical probability.
Subjective Probability: Probability that represents someone's personal degree of belief.
Probability Assignment Rule
The sum of the probabilities of all possible outcomes in the sample space must be 1.
Formula: $P(S) = 1$
Complement Rule
The probability that an event does not occur is 1 minus the probability that it does occur.
Formula: $P(A^c) = 1 - P(A)$
Example: If the probability of rain tomorrow is 0.3, then the probability it does not rain is $1 - 0.3 = 0.7$.
Disjoint (Mutually Exclusive) Events
Two events are disjoint (mutually exclusive) if they have no outcomes in common. If A and B are disjoint, knowing that A occurs tells us that B cannot occur.
Example: Rolling a die: the events "rolling a 2" and "rolling a 5" are disjoint.
Addition Rule for Disjoint Events
If A and B are disjoint events, the probability that A or B occurs is the sum of their probabilities.
Formula: $P(A \cup B) = P(A) + P(B)$
Example: If $P(A) = 0.2$ and $P(B) = 0.3$, then $P(A \cup B) = 0.5$.
Legitimate Probability Assignment
Assigning probabilities to outcomes is legitimate if:
Each probability is between 0 and 1 (inclusive).
The sum of all probabilities is 1.
Multiplication Rule for Independent Events
If A and B are independent events, the probability that both A and B occur is the product of their probabilities.
Formula: $P(A \cap B) = P(A) \times P(B)$
Example: If $P(A) = 0.4$ and $P(B) = 0.5$, then $P(A \cap B) = 0.2$.
Independence Assumption
We often assume events to be independent when there is no reason to think otherwise. Independence means the occurrence of one event does not affect the probability of the other.
Example: Tossing two coins: the result of one toss does not affect the other.
