Glossary · Topic 6

L01 Simple probability. Open Lesson 1 →

Experiment: Any process with an uncertain outcome: rolling a die, flipping a coin, drawing a card, spinning a wheel. The probability questions in this topic are about experiments.
Outcome: A single possible result of the experiment. Rolling a die has 6 outcomes (1, 2, 3, 4, 5, 6). Drawing a card has 52.
Sample space: The set of all possible outcomes. For a die, the sample space is {1, 2, 3, 4, 5, 6}. For a card draw, the sample space has 52 elements.
Event: A subset of the sample space — the outcomes we're interested in counting as "yes." "Rolling an even number" is the event {2, 4, 6} inside the die's sample space.
Equally likely: Every outcome in the sample space has the same chance of occurring. This lesson assumes equally-likely outcomes (fair die, well-shuffled deck). When that assumption fails, the simple-ratio formula doesn't apply directly — you weight by individual outcome probabilities instead (Lesson 5).
Probability range: Every probability is a number between 0 and 1 (inclusive). 0 means impossible, 1 means certain, and everything in between is the rest of life. If you compute a probability that comes out negative or above 1, you made an arithmetic error somewhere.

L02 Fundamental counting principle. Open Lesson 2 →

Independent steps: Steps where the choice at one step does not constrain the choices at any other step. License-plate slots are independent (letter A in slot 1 doesn't block letter A in slot 2); medals in a race are not (the gold medalist can't also win silver).
With replacement / repeats allowed: After making a selection at step k, that choice is still available at step k+1. License plates allow repeats. A bag where you draw a ball, record it, and put it back is "with replacement."
Without replacement / no repeats: Once a choice is made, it can't be made again. Medals (gold, silver, bronze) can't go to the same person twice. The number of choices shrinks at each step (n, n−1, n−2, ...). That setup is a permutation, introduced in L4.
Tree diagram: A visual organizer for multi-step problems with few choices. Each step adds a layer of branches; the total leaves at the bottom equals the product of the branches per step. Useful for sample-space problems with 2-3 steps and small choice counts.
Shrinking choices: In a multi-step count where repeats aren't allowed, the number of choices drops by 1 at each step. "Pick 3 books from 10 with no repeats" = 10 × 9 × 8. The pattern of descending integers is the seed of the factorial.

L03 Sample spaces and the addition rule. Open Lesson 3 →

Union (A or B): The event "A happens OR B happens (or both)." Includes every outcome that's in either set. In set notation: A ∪ B.
Intersection (A and B): The event "A happens AND B happens at the same time." Includes only the overlap. In set notation: A ∩ B. The intersection is what the addition rule subtracts.
Mutually exclusive: Two events are mutually exclusive when they cannot both occur on the same trial. Rolling a 3 and rolling a 5 on the same die roll. Their intersection is empty, so P(A and B) = 0 and the addition rule simplifies.
Standard 52-card deck: 52 cards in 4 suits (♠ ♣ ♥ ♦), 13 cards per suit (A, 2-10, J, Q, K). Hearts and diamonds are red (26 cards); spades and clubs are black (26 cards). Face cards are J, Q, K (12 total, three per suit). The canonical deck shows up in every ALEKS probability review.
Double-counting: The mistake of including the same outcome twice. Happens when you compute P(A or B) as P(A) + P(B) without subtracting the overlap. Always subtract P(A and B) — the inclusion-exclusion rule is the fix.

L04 Permutations and combinations. Open Lesson 4 →

Factorial (n!): n! = n × (n−1) × (n−2) × ··· × 1. So 3! = 6, 4! = 24, 5! = 120, 10! = 3,628,800. By convention, 0! = 1. The factorial counts the number of ways to arrange n distinct items in a row.
P(n, r) — permutation: Number of ordered ways to pick r items from n: P(n, r) = n × (n−1) × ··· × (n−r+1), which is r terms total. Equivalently, n!/(n−r)!. Use when the position or identity of each pick matters.
C(n, r) — combination: Number of unordered ways to pick r items from n: C(n, r) = P(n, r) / r!. Use when the order or position doesn't matter and every selection of the same r items is the same answer.
The litmus test: "Does swapping two of my picks give me a different answer?" If yes (medals, schedules, podium positions), it's a permutation. If no (committees, card hands, pizza toppings), it's a combination. Ask this question first, before reaching for any formula.
Shrinking product: The descending multiplication n × (n−1) × (n−2) × … that shows up in every permutation. P(10,3) = 10 × 9 × 8 = 720. Stop after r factors. The factorial is just the shrinking product all the way down to 1.

L05 Expected value. Open Lesson 5 →

Random variable (X): A quantity whose value depends on the outcome of a random experiment. The payoff on a spinner spin, the number of heads in 10 coin flips, the score on a die roll. Usually written X (or another capital letter).
Expected value (E(X)): The mean (average) of the random variable, weighted by probability: Σ x · P(x). Also written μ in some textbooks. Has the same units as the random variable itself (dollars per spin, points per roll, etc.).
Fair game: A game with E(X) = 0. Neither player nor house has a long-run advantage. Pure fair games are rare in practice — most casino games have a slightly negative E for the player (the "house edge") and insurance premiums are set to give the insurance company a slightly positive E.
Long-run interpretation: E(X) is not what you should expect to win on any single trial — it's the average across many trials. If E(X) = $0.50, you should expect to be ahead by about $50 after 100 plays, $500 after 1,000 plays. Variance still matters in the short run.
House edge: When the operator of a game (a casino, a lottery, an insurance company) has a negative expected value from the player's side, the absolute value of that E(X) is the house edge. Roulette's edge is about $0.053 lost per dollar bet, every spin, on average. That's the long-run profit per trial that keeps the lights on.

L06 Theoretical vs experimental probability. Open Lesson 6 →

Compound event: An event built from two or more simpler events (typically with AND or OR). "The ball is odd AND the card is K or J" is a compound event made from two simple ones.
Independent events: Two events are independent when the occurrence of one does not change the probability of the other. Drawing a card from one deck and rolling a die are independent. For independent A and B: P(A and B) = P(A) · P(B).
Law of Large Numbers: As the number of trials of a random experiment grows, the experimental probability of an event approaches the theoretical probability. Short runs are noisy; long runs are not. This is why casinos make consistent profits despite any individual player winning big — the house plays the long run.
Convergence (not equality): The Law of Large Numbers says experimental P gets closer to theoretical P as n grows, not that they become equal. Even at n = 1,000,000, the experimental value will differ slightly from the theoretical — just by a smaller amount than at n = 100.
Experimental vs. theoretical: Two ways to assign a probability. Experimental: count what actually happened in the data — favorable trials ÷ total trials. Theoretical: compute from the structure of the experiment — favorable outcomes ÷ total outcomes. Both are valid; they should agree as the sample grows.

Vocabulary & key terms