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Ch 6: Probability·§6.0 Chapter overview
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Chapter 6

Probability

Counting, permutations and combinations, theoretical and experimental probability, and expected value. The math of uncertainty — and the engine behind every insurance premium, lottery ticket, and game of chance.

This chapter is probability — the math of uncertainty. By the end you'll be fluent in the probability ratio (favorable / total), the fundamental counting principle that powers every "how many possible outcomes" question, permutations vs combinations (does order matter?), and the expected value machine that puts a fair price on any game of chance.

Chapter 6 is the second half of the data side of the course. Chapter 5 was descriptive — how to read a data set you already have. Chapter 6 is predictive — what to expect from a process whose outcome isn't decided yet. Together they give you the tools to read polls, studies, and games of chance with the same fluency a statistician has.

By the end of this chapter, you'll be able to…

  • 6.1Compare, compute, and interpret theoretical and empirical probabilities.
  • 6.2Develop and use the fundamental counting principle.
  • 6.3Develop a procedure for finding expected value.
  • 6.4Compute and interpret expected values.

Sections

  1. 6.1What probability is, and the favorable-over-total ratioProbability is a number between 0 and 1 that measures how likely an event is to occur. For simple events with equally-likely outcomes, the math reduces to one ratio: favorable outcomes over total outcomes.Read →
  2. 6.2The fundamental counting principleWhen a process has several independent steps, the total number of possible outcomes is the product of the number of choices at each step. The single most-used move in the topic.Read →
  3. 6.3Sample spaces and the addition ruleP(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap once prevents counting the shared outcomes twice. The standard 52-card deck is the canonical training ground.Read →
  4. 6.4Permutations and combinationsWhen choices are made without replacement, the counting principle yields shrinking products. Permutations count ordered arrangements; combinations count unordered selections; the two differ by a factor of r!.Read →
  5. 6.5Expected valueE(X) = Σ x · P(x). The single number that summarizes the long-run average payout of a random process. The math of fair games, insurance premiums, and casino edges.Read →
  6. 6.6Theoretical and experimental probabilityTheoretical probability comes from the structure of the experiment; experimental probability comes from the data of running it. The Law of Large Numbers says that as trials accumulate, the two converge.Read →
Start with §6.1 →

Chapter glossary

All key terms introduced across this chapter, in the order they appear in the reading.

Experiment
Any process with an uncertain outcome — rolling a die, drawing a card, spinning a wheel, flipping a coin.
Outcome
A single possible result of an experiment. A die roll has six outcomes; a card draw has 52.
Sample space
The set of all possible outcomes of an experiment.
Event
A subset of the sample space — the outcomes counted as "yes" for a given question.
Probability (theoretical)
For equally-likely outcomes, P(A) = n(A) / n(S), the count of favorable outcomes divided by the total. Always in [0, 1].
Complement
The event Ac consisting of every outcome not in A. P(Ac) = 1 − P(A).
Fundamental counting principle
For a multi-step process with independent steps, the total number of outcomes is the product of the number of choices at each step.
Union (A or B)
The event that A or B (or both) occurs. A ∪ B.
Intersection (A and B)
The event that A and B both occur. A ∩ B.
Mutually exclusive
Two events that cannot both occur on the same trial. Their intersection is empty, so P(A ∩ B) = 0.
Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the overlap once to avoid double-counting.
Independent events
Two events whose occurrence does not affect each other. For independent A and B, P(A ∩ B) = P(A) · P(B).
Factorial (n!)
The product n · (n−1) · ... · 1. Counts the number of orderings of n distinct objects. 0! = 1 by convention.
Permutation P(n, r)
Number of ordered selections of r items from n distinct items. P(n, r) = n! / (n − r)!.
Combination C(n, r)
Number of unordered selections of r items from n distinct items. C(n, r) = n! / [(n − r)! · r!].
Random variable
A numerical quantity whose value depends on the outcome of a random experiment.
Expected value E(X)
The probability-weighted average of a random variable: Σ x · P(x). The long-run average per trial.
Fair game
A game with E(X) = 0; neither player nor opponent has a long-run advantage.
Experimental probability
The proportion of trials, in a sequence of n repetitions of an experiment, on which the event occurred. Depends on the data, not the structure.
Law of Large Numbers
As the number of trials grows, the experimental probability of an event converges to the theoretical probability. Convergence, not equality.
Standard 52-card deck
Four suits (spades, clubs, hearts, diamonds), 13 ranks each. Hearts and diamonds are red; spades and clubs are black. Face cards are the J, Q, K (12 total).
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