Fundamental Probability Concepts and Rules

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, each with its own context and application.

• Theoretical Probability: When probability is based on a model with equally likely outcomes, it is called theoretical probability. Example: The probability of rolling a 3 on a fair six-sided die is $\frac{1}{6}$.

• Subjective Probability: Probability that represents someone's personal degree of belief. Example: Estimating the chance of rain tomorrow based on personal judgment.

Probability Assignment Rule

The sum of the probabilities of all possible outcomes in the sample space must be 1.

• $P(S) = 1$ where $S$ is the sample space.

Probability Rules

• Complement Rule: The probability that an event does not occur is 1 minus the probability that it does occur. $P(A^c) = 1 - P(A)$ Example: If the probability of rain today is 0.3, then the probability it does not rain is $1 - 0.3 = 0.7$.

• Disjoint (Mutually Exclusive) Events: Two events are disjoint if they have no outcomes in common. If A and B are disjoint, knowing that A occurs tells us that B cannot occur. Example: Drawing a card that is both a heart and a spade from a deck is impossible; these events are disjoint.

• Addition Rule: If A and B are disjoint events, the probability that A or B occurs is: $P(A \cup B) = P(A) + P(B)$

• Legitimate Probability Assignment:

  • Each probability is between 0 and 1 (inclusive).

  • The sum of all probabilities is 1.

• Multiplication Rule: If A and B are independent events, the probability that both A and B occur is: $P(A \cap B) = P(A) \times P(B)$ Example: The probability of flipping two heads in two coin tosses is $0.5 \times 0.5 = 0.25$.

• Independence Assumption: Events are independent if the occurrence of one does not affect the probability of the other. This assumption should be checked for reasonableness in context.

Summary Table: Probability Rules

Rule	Formula	When to Use
Complement Rule	$P(A^c) = 1 - P(A)$	Finding the probability that an event does not occur
Addition Rule (Disjoint Events)	$P(A \cup B) = P(A) + P(B)$	Events cannot occur together
Multiplication Rule (Independent Events)	$P(A \cap B) = P(A) \times P(B)$	Events do not affect each other