Understanding Probability Basics

The Basic Definition of Probability

The probability of an event $A$ occurring is given by:

This formula tells us how many ways our desired event can happen compared to all possible outcomes. Probability always ranges from 0 (impossible) to 1 (certain).

Understanding Sample Space and Events

• Sample space - all possible outcomes of an experiment;
• Event - a specific outcome or set of outcomes we're interested in.

Example with flipping a coin:

• Sample space = {Heads, Tails} ;
• Event A = {Heads} .

Then:

Union Rule: "A OR B Happens"

Definition: the union of two events $A \cup B$ represents outcomes where either $A$ occurs, or $B$ occurs, or both occur.

Formula:

We subtract the intersection to avoid double-counting outcomes that appear in both events.

Union Example: Rolling a Die

Let's roll a six-sided die:

• Event A = {1, 2, 3} (rolling a small number)
• Event B = {2, 4, 6} (rolling an even number)

Union and intersection:

• $A \cup B = \{1, 2, 3, 4, 6\}$
• $A \cap B = \{2\}$

Calculations step-by-step:

Apply the union formula:

Intersection Rule: "A AND B Both Happen"

Definition: The intersection of two events $A \cap B$ represents outcomes where both $A$ and $B$ occur simultaneously.

General Formula

In all cases:

where $P(B|A)$ is the conditional probability of $B$ given that $A$ has already occurred.

Case 1: Independent Events

If the events do not affect each other (e.g., flipping a coin and rolling a die):

Example:

• $P(\text{Head on a coin}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$2$}}$;
• $P(\text{6 on a die}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$6$}}$.

Then:

Case 2: Dependent Events

If the result of the first event influences the second (e.g., drawing cards without replacement):

Example:

• $P(\text{first card is an Ace}) = \tfrac{\raisebox{1pt}{$4$}}{52}$;
• $P(\text{second card is an Ace | first card was an Ace}) = \tfrac{\raisebox{1pt}{$3$}}{\raisebox{-1pt}{$51$}}$.

Then: