Probability Fundamentals

Probability is the mathematics of uncertainty. It gives us a precise language for reasoning about chance events -- from card games to weather forecasting to medical testing. Every probability is a number between 0 (impossible) and 1 (certain).

Core Definitions

Term	Definition	Example
Experiment	A process with uncertain outcomes	Rolling a die
Sample space (S)	The set of all possible outcomes	S = {1, 2, 3, 4, 5, 6}
Event (A)	A subset of the sample space	A = "rolling even" = {2, 4, 6}
P(A)	Probability of event A	P(A) = 3/6 = 1/2

P(A) = number of favorable outcomes / total number of outcomes

(This formula applies when all outcomes are equally likely.)

Fundamental Rules

Complement Rule

P(not A) = 1 - P(A)

The probability something does NOT happen equals 1 minus the probability it does happen.

Addition Rule (OR)

P(A or B) = P(A) + P(B) - P(A and B)

We subtract P(A and B) to avoid double-counting outcomes in both events.

If A and B are mutually exclusive (cannot happen simultaneously), then P(A and B) = 0:

P(A or B) = P(A) + P(B) (mutually exclusive events)

Multiplication Rule (AND)

P(A and B) = P(A) \times P(B|A)

If A and B are independent (one does not affect the other), then P(B|A) = P(B):

P(A and B) = P(A) \times P(B) (independent events)

Independent vs. Dependent Events

Independent: flipping a coin twice -- the second flip is not affected by the first.

Dependent: drawing two cards without replacement -- the second draw depends on the first.

Counting Principle

If one task can be done in m ways and a second task in n ways, the two tasks together can be done in m × n ways.

Worked Example 1 -- Basic Probability

A bag contains 4 red, 3 blue, and 5 green marbles. What is P(blue)?

Total marbles = 4 + 3 + 5 = 12.
P(blue) = 3/12 = 1/4 = 0.25.

Worked Example 2 -- Addition Rule

In a standard deck of 52 cards, what is P(heart or face card)?

P(heart) = 13/52. P(face card) = 12/52. P(heart AND face card) = 3/52 (J, Q, K of hearts).
P(heart or face card) = 13/52 + 12/52 - 3/52 = 22/52 = 11/26 ≈ 0.423.

Worked Example 3 -- Independent Events

A fair coin is flipped 3 times. What is P(all heads)?

Each flip is independent with P(H) = 1/2.
P(HHH) = 1/2 × 1/2 × 1/2 = 1/8 = 0.125.

Common Mistake

Adding probabilities for "or" without subtracting the overlap. If events can happen simultaneously, you must use P(A or B) = P(A) + P(B) - P(A and B). Skipping the subtraction leads to a probability that may exceed 1 -- which is always wrong.

Practice Problems

A die is rolled. What is P(number greater than 4)?
Show Solution

Favorable outcomes: {5, 6}. P = 2/6 = 1/3.
Two dice are rolled. How many outcomes are in the sample space?
Show Solution

By the counting principle: 6 × 6 = 36 outcomes.
P(rain tomorrow) = 0.3. What is P(no rain tomorrow)?
Show Solution

P(no rain) = 1 - 0.3 = 0.7.
A bag has 5 red and 7 blue marbles. You draw 2 without replacement. What is P(both red)?
Show Solution

P(1st red) = 5/12. P(2nd red | 1st red) = 4/11. P(both red) = (5/12)(4/11) = 20/132 = 5/33 ≈ 0.152.
A password is 3 digits (0-9). How many possible passwords are there? What is the probability of guessing correctly on one try?
Show Solution

10 × 10 × 10 = 1,000 passwords. P(correct) = 1/1000 = 0.001.

Summary

Probability ranges from 0 to 1: P(A) = favorable / total (equally likely outcomes).
Complement rule: P(not A) = 1 - P(A).
Addition rule: P(A or B) = P(A) + P(B) - P(A and B).
Multiplication rule: P(A and B) = P(A) × P(B) for independent events.
Counting principle: m × n total ways for sequential tasks.