Unit 4: Probability, Random Variables, and Probability Distributions

Probability (long-run model)

A way to model random processes by describing how often outcomes occur in the long run, not by predicting individual results.

Random process

A process where individual outcomes can’t be predicted with certainty, but long-run patterns (regularity) can be predictable.

Relative frequency

The proportion of times an event occurs in repeated trials; used as an estimate of probability.

Law of Large Numbers (LLN)

As an experiment is repeated many times (with an unchanging chance process), the relative frequency of an event tends to get closer to its true probability.

Short-run variability (streaks)

In the short run, random results can look unusual (including streaks), even when the long-run proportion is stable.

Gambler’s fallacy

The mistaken belief that after a long streak, the opposite outcome is “due”; random processes do not have memory in that way (e.g., a fair coin stays 0.5 for heads each flip).

Simulation

A method that imitates a random process (using random digits, random number generators, or physical randomizers) to estimate probabilities when exact calculations are difficult.

Simulation setup

The planning stage of a simulation: define outcomes, assign random numbers/digits to outcomes with the correct probabilities, and specify what counts as “success” in each trial.

Simulation trial

One run of the simulated chance process that mimics one repetition of the real situation (e.g., reading three two-digit numbers to represent three free throws).

Simulation estimate

An estimated probability found by repeating many simulation trials and using the proportion of trials in which the event occurs.

Sample space

The set of all possible outcomes of a random process (e.g., {1,2,3,4,5,6} for one die roll).

Event

A subset of the sample space; a collection of outcomes of interest (e.g., “roll an even number” = {2,4,6}).

Equally likely outcomes

A model assumption where each outcome in the sample space has the same probability, allowing P(A) = (outcomes in A)/(outcomes in sample space).

Complement rule

For an event A, the complement A^c is “A does not happen,” and P(A^c) = 1 − P(A).

Union (inclusive OR)

A ∪ B means “A or B or both” occur (inclusive OR).

Intersection (AND)

A ∩ B means both A and B occur.

Mutually exclusive (disjoint) events

Events that cannot happen at the same time; P(A ∩ B) = 0.

Addition rule (general)

For any events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Multiplication rule (general)

For any events A and B: P(A ∩ B) = P(A)P(B|A) (equivalently, P(B)P(A|B)).

Complement strategy (“at least one”)

Compute “at least one” by subtracting the complement: P(at least one) = 1 − P(none).

Conditional probability

P(A|B) is the probability A occurs given B occurred: P(A|B) = P(A ∩ B)/P(B), for P(B) > 0.

Independence

Events A and B are independent if knowing one occurred does not change the probability of the other (e.g., P(A|B)=P(A) or P(A∩B)=P(A)P(B)).

Mutually exclusive vs. independent

Mutually exclusive events (with positive probabilities) cannot be independent because their intersection probability is 0, not P(A)P(B).

Joint probability

The probability two events happen together, written as an intersection (e.g., P(A ∩ B)).

Two-way table

A table of counts (or proportions) for two categorical variables; used to compute marginal, joint, and conditional probabilities clearly.

Conditional probability from a two-way table

A conditional probability computed using the appropriate row/column total as the denominator, determined by the “given” condition.

Tree diagram

A diagram for multistage probability where branches represent outcomes with conditional probabilities at each stage.

Total probability with a tree

Multiply probabilities along each path for sequence probabilities, then add path probabilities to find an overall probability (e.g., sum across groups).

Bayes’ rule

A method to reverse a conditional probability: P(A|B) = [P(B|A)P(A)]/P(B), often implemented more safely using a table of counts.

Base-rate effect

When the underlying rate of a condition is low, even a good test can produce many false positives; the prior probability heavily influences P(condition|positive).

Random variable

A numerical variable whose values come from a random process (turns outcomes into numbers for analysis).

Discrete random variable

A random variable that takes a countable set of possible values (e.g., 0,1,2,… or 0–5).

Probability distribution (discrete)

A list or rule giving P(X=x) for each possible value x of a discrete random variable X.

Valid probability distribution criteria

A discrete distribution is valid if every probability is between 0 and 1 and the total of all probabilities sums to 1 (within rounding).

Cumulative distribution function (CDF)

A function/table giving cumulative probabilities such as P(X ≤ x), used for “at most,” “no more than,” and “less than or equal to” questions.

Expected value E(X) (mean)

The long-run average value of a random variable: μ = E(X) = Σ x·P(X=x).

Expected net gain / fair game

Using expected value to judge long-run profitability/fairness; a game is fair when expected net gain is 0 (positive favors the player, negative favors the house).

Variance and standard deviation (discrete)

Measures of spread for a discrete random variable: variance σ² = Σ (x−μ)²P(X=x) and standard deviation σ = √σ².

Variance shortcut

An equivalent formula: Var(X)=σ²=E(X²)−(E(X))², where E(X²)=Σ x²P(X=x).

z-score

A standardized value measuring how many standard deviations an observation is from the mean: z = (value − mean)/SD; large |z| indicates a surprising outcome.

Linear transformation Y=a+bX

A transformation of a random variable where μY = a + bμX and σY = |b|σX (adding a constant shifts center; multiplying scales spread).

Mean of sums/differences

For random variables X and Y: μ{X+Y}=μX+μY and μ{X−Y}=μX−μY (means always add/subtract).

Variance of sums/differences (independent case)

If X and Y are independent: Var(X+Y)=Var(X−Y)=Var(X)+Var(Y); add variances (not standard deviations).

Binomial distribution (Bin(n,p), BINS)

Models the number of successes in a fixed number n of independent trials with binary outcomes and constant success probability p (BINS: Binary, Independent, Number fixed, Same p).

Binomial probability formula

For X ~ Bin(n,p), the probability of exactly k successes is P(X=k)=C(n,k)p^k(1−p)^{n−k}.

Binomial mean and standard deviation

For X ~ Bin(n,p): μ=np and σ=√(np(1−p)).

10% condition (binomial with sampling)

When sampling without replacement, a binomial model is often acceptable if the sample size is no more than 10% of the population, making independence reasonable.

Geometric distribution (Geometric(p))

Models the number of trials until the first success (including the success trial) for independent trials with constant success probability p; mean μ=1/p.

Geometric probability formula

For X ~ Geometric(p): P(X=k)=(1−p)^{k−1}p for k=1,2,3,… (the exponent k−1 counts failures).

Memoryless property (geometric)

For a geometric setting, past failures do not change future success probability; after many failures, the chance of success on the next trial is still p.