Probability

Probability is the mathematics of uncertainty. When you flip a coin, roll a die, or make decisions with incomplete information, probability quantifies how likely outcomes are. Rather than guessing, probability gives you a numerical measure from 0 (impossible) to 1 (certain). Understanding probability is essential for risk assessment, decision-making, gambling, insurance, weather forecasting, medical diagnosis, and scientific inference. This chapter introduces the theoretical approach to probability—calculating likelihood from first principles using equally likely outcomes.

The Sample Space: All Possible Outcomes

Before calculating probability, you must identify all possible outcomes.

Sample space (S): The set of all possible outcomes of an experiment.

Examples:

• Coin flip: S = {Head, Tail} (2 outcomes)
• Single die roll: S = {1, 2, 3, 4, 5, 6} (6 outcomes)
• Two coin flips: S = {HH, HT, TH, TT} (4 outcomes)
• Selecting a card: S = {all 52 cards} (52 outcomes)

Key insight: Each outcome in a sample space must be equally likely (for the definition of probability to apply directly). A fair coin has equally likely outcomes (each 50%). A biased coin doesn't.

Events: Subsets of the Sample Space

An event (E) is a subset of the sample space—a collection of outcomes you're interested in.

Examples:

• Coin flip: "Getting a Head" = {Head} (1 outcome out of 2)
• Die roll: "Getting an even number" = {2, 4, 6} (3 outcomes out of 6)
• Two die rolls: "Sum equals 7" = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} (6 outcomes out of 36)

An event can be:

• Certain (the entire sample space): probability = 1
• Impossible (the empty set): probability = 0
• Partial (some outcomes): probability between 0 and 1

The Theoretical Probability Formula

For equally likely outcomes:

P(Event E) = (Number of favorable outcomes) / (Total number of outcomes)
           = |E| / |S|

Where |E| means the count of outcomes in event E, and |S| means the count of all outcomes.

Examples:

• Probability of rolling a 3: P = 1/6 ≈ 0.167 or 16.7%
• Probability of rolling even: P = 3/6 = 1/2 = 0.5 or 50%
• Probability of rolling 1 or 2: P = 2/6 = 1/3 ≈ 0.333 or 33.3%

This formula assumes:

1. All outcomes are equally likely
2. Outcomes are mutually exclusive (can't happen simultaneously)
3. The sample space is complete (includes all possibilities)

Complementary Events: The "Not" Relationship

The complement of event E is "not E"—all outcomes where E doesn't occur.

P(E) + P(not E) = 1

Or equivalently:

P(not E) = 1 − P(E)

Intuition: You either get E or you don't. The probabilities must sum to 1.

Example:

• P(rolling a 3) = 1/6
• P(not rolling a 3) = 1 − 1/6 = 5/6

Sometimes calculating P(not E) is easier than P(E) directly. If you want "at least one success," it's often easier to calculate "no successes" and subtract from 1.

Multiple Events: Combined Probabilities

When dealing with multiple events, you need careful language:

"Or" (union): Event A or Event B (or both) occur

• Depends on whether events overlap (are mutually exclusive)
• For mutually exclusive: P(A or B) = P(A) + P(B)

"And" (intersection): Event A and Event B both occur

• For independent events: P(A and B) = P(A) × P(B)
• (Independence means one event's outcome doesn't affect the other)

Example with a die:

• P(rolling 1 or 2) = P(1) + P(2) = 1/6 + 1/6 = 1/3 (mutually exclusive)
• P(rolling even on die 1 AND rolling odd on die 2) = 1/2 × 1/2 = 1/4 (independent)

Real-World Applications: From Theory to Decision-Making

Insurance: Probability of accidents determines premium costs

Weather: Probability of rain affects planning

Medical diagnosis: Probability that a positive test means you have the disease (more complex than the basic formula)

Lottery: Astronomically small probabilities explain why lotteries aren't profitable for players

Quality control: Probability of defects determines batch acceptance

Connecting to Related Topics

Probability builds on and relates to:

• chapter-13-statistics: Statistics observes actual outcomes; probability predicts theoretical ones
• chapter-01-real-numbers: Counting outcomes uses divisibility and number properties
• chapter-05-arithmetic-progressions: Some probability problems involve sequences

Key Formulas and Concepts

• Theoretical probability: P(E) = (favorable outcomes) / (total outcomes)
• Complement rule: P(not E) = 1 − P(E)
• Mutually exclusive events: P(A or B) = P(A) + P(B)
• Independent events: P(A and B) = P(A) × P(B)
• Sample space: The set of all possible outcomes
• Favorable outcome: An outcome in event E

Socratic Questions

1. If you flip two fair coins, why are there 4 equally likely outcomes (HH, HT, TH, TT) rather than 3 outcomes (2 heads, 1 head, 0 heads)?

1. When rolling two dice, there are 36 equally likely outcomes. But when counting the sum (2 through 12), the outcomes are NOT equally likely. Why is the sum not uniformly distributed?

1. If the probability of an event is 0.3, what is the probability that it doesn't occur? Can you explain why these must sum to 1?

1. Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what is P(A and B)? What if they were mutually exclusive instead?

1. In a game, you win if you roll a die and get 5 or 6 on the first roll, OR you roll a 6 on the second roll. How would you calculate your probability of winning?