Unit 2: Probability, Random Variables, and Probability Distributions

Two Categorical Variables

When two categorical variables are measured on the same individuals, organize the data in a two-way table. Each cell records the count for one combination of categories.

	Category 1	Category 2	Total
Group A	$a$	$b$	$a+b$
Group B	$c$	$d$	$c+d$
Total	$a+c$	$b+d$	$n$

Marginal relative frequencies use row or column totals to describe one variable by itself. Conditional relative frequencies restrict attention to one row or column and describe the distribution of the other variable within that condition.

Simulation

A simulation uses random digits, technology, cards, dice, or another chance device to imitate a random process. Simulations are useful when exact probability is hard to calculate or when you want to understand long-run behavior before using formulas.

Good simulations clearly define:

What one trial represents.
How random outcomes are assigned to the real-world outcomes.
What statistic is recorded after each trial.
How many repetitions are used.
How the simulation results answer the original question.

Sample space and experiments

A random phenomenon or probability experiment is a process with outcomes that vary from trial to trial in a way that cannot be predicted with certainty in advance, but whose possible outcomes are known.
The sample space, denoted $S$ , is the set of all possible outcomes of that experiment. An event is any subset of the sample space (a collection of one or more outcomes). The letter $S$ is standard notation; individual outcomes are often written as simple labels or ordered pairs when the experiment has multiple stages.
A tree diagram lists stages of an experiment as branches. Multiply along a path to get the probability of that path when stages are suitably independent or conditional probabilities are marked on branches; add paths that represent the same event. Tree diagrams keep ordered outcomes visible and help avoid double-counting when the experiment is multistep.

Basic rules of probability

Rule 1 (bounds): For any event $A$ ,

0 \le P(A) \le 1

Rule 2 (whole sample space): If each possible outcome is listed exactly once, the sum of their probabilities is $1$ .
Rule 3 (impossibilility): An impossible event has probability $0$ . A sure event (the entire sample space, or any event that must happen) has probability $1$ .
Rule 4 (odds in favor): Odds in favor of an event $A$ compare the chance $A$ occurs to the chance it does not. With complement $A'$ (or $A^C$ ) for “not $A$ ,”

\text{Odds in favor of } A = \frac{P(A)}{P(A')}

Odds are a ratio, not a probability; you can recover probabilities from odds with a little algebra when needed.

Events: complement, disjoint, union, intersection

The complement of an event $A$ is the event that $A$ does not occur. It includes every outcome in $S$ that is not in $A$ . Notation includes $A'$ and $A^C$ .
Disjoint events (also called mutually exclusive events) cannot both occur on the same trial: they share no outcomes, so $P(A \cap B) = 0$ when $A$ and $B$ are disjoint.
The union $A \cup B$ is “ $A$ or $B$ or both”—at least one of the two events happens.
The intersection $A \cap B$ is “ $A$ and $B$ both” happen.
Conditional probability “ $A$ given $B$ ” restricts attention to outcomes where $B$ occurred. The notation is

A \mid B

read as “ $A$ given $B$ .”

Two events are independent if knowing whether one occurred does not change the probability of the other (formalized below).

More probability rules

Complement rule:

P(A') = 1 - P(A)

General addition rule (union):

P(A \cup B) = P(A) + P(B) - P(A \cap B)

If $A$ and $B$ are disjoint, then $P(A \cap B) = 0$ and the rule reduces to $P(A \cup B) = P(A) + P(B)$ .

General multiplication rule (intersection):

P(A \cap B) = P(A)\,P(B \mid A) = P(B)\,P(A \mid B)

From the multiplication rule you obtain the standard formula for conditional probability—the same rearrangement that describes Bayes’ theorem in tree-and-table problems:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

provided $P(B) > 0$ .

Independence (equivalent formulations for events with positive probability):

P(A \mid B) = P(A) \quad \text{and} \quad P(B \mid A) = P(B)

Equivalently, independence is often written as

P(A \cap B) = P(A)\,P(B)

If that product rule fails, the events are dependent.

Random variables and probability distributions

A random variable assigns a numerical value to each outcome of a random experiment. Customarily we use capital letters such as $X$ or $Y$ for the variable and lowercase $x$ for a possible value it might take.

A discrete random variable takes a finite or countably infinite set of values (counts, “number of successes,” and so on).
A continuous random variable takes values in an interval (time, weight, distance). Probabilities for continuous models are assigned to intervals using density and area ideas in later work; this unit focuses on the discrete case.

Discrete probability distributions

A discrete probability distribution lists every possible value $x_i$ of $X$ together with its probability $P(x_i)$ (or $P(X = x_i)$ ). The list may appear as a table, a formula, or a probability histogram.

Let $X$ take values $x_1, x_2, \ldots, x_n$ with probabilities $P(x_1), P(x_2), \ldots, P(x_n)$ . The pairing

\{(x_1, P(x_1)), (x_2, P(x_2)), \ldots, (x_n, P(x_n))\}

is a valid probability distribution if

0 \le P(x_i) \le 1 \quad \text{for all } i = 1, 2, \ldots, n

and

\sum_{i=1}^{n} P(x_i) = 1

The first condition keeps each entry a legitimate probability; the second says exactly one of the listed values occurs (for a complete list of possibilities).

Mean of a discrete random variable

The expected value $E(X)$ of a discrete random variable $X$ is also called its mean and denoted $\mu_X$ when emphasis is on the distribution. It is the probability-weighted average of the possible values:

\mu = E(X) = \sum_{i=1}^{n} x_i\, P(x_i)

That number is a center of the distribution, but it need not be a value $X$ can actually take.

Bonus!

Surprisingly, expected value is an additive property. The linearity of expectation states that:

E(X_1 + X_2 + ... + X_n) = E(X_1) + E(X_2) + ... + E(X_n)

This property is useful in games, insurance, and counting problems because it does not require the random variables to be independent. Variance rules, however, do require independence in the simple forms used in AP Statistics.

Variance of a discrete random variable

The variance $\sigma^2$ (or $\operatorname{Var}(X)$ ) measures spread around the mean as a probability-weighted average of squared deviations:

\sigma^2 = \sum_{i=1}^{n} (x_i - \mu)^2\, P(x_i)

The standard deviation is $\sigma = \sqrt{\sigma^2}$ , returned to the original units of $X$ .

Combinations (binomial coefficient)

A combination counts how many ways you can choose $r$ objects from $n$ distinct objects when order does not matter. The symbol is the binomial coefficient $\binom{n}{r}$ , read “ $n$ choose $r$ ”:

\binom{n}{r} = \frac{n!}{r!\,(n-r)!}

This expression appears in the binomial probability model (fixed $n$ independent trials, same success probability $p$ , count successes) and in many counting-based probability problems on the exam.

Binomial and geometric models

A binomial random variable counts successes in a fixed number of independent trials with the same success probability. If $X \sim \operatorname{Binomial}(n,p)$ , then

P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}.

The mean and standard deviation are

\mu_X=np,\qquad \sigma_X=\sqrt{np(1-p)}.

A geometric random variable counts trials until the first success. If $X \sim \operatorname{Geometric}(p)$ , then

P(X=k)=(1-p)^{k-1}p.

Normal Distributions

A normal distribution is a symmetric, bell-shaped distribution described by its mean $\mu$ and standard deviation $\sigma$ :

X \sim N(\mu,\sigma).

To standardize a value,

z=\frac{x-\mu}{\sigma}.

Sampling Distributions and the Central Limit Theorem

A sampling distribution is the distribution of a statistic over many random samples of the same size from the same population. It describes how a statistic varies from sample to sample.

For a statistic, pay attention to:

Shape: normal, skewed, approximately symmetric, etc.
Center: the mean of the statistic.
Spread: the standard deviation of the statistic, often called the standard error.

For a sample mean,

\mu_{\bar{x}}=\mu,\qquad \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}.

The Central Limit Theorem says that when $n$ is large, the sampling distribution of $\bar{x}$ is approximately normal, even if the population distribution is not normal, as long as observations are independent and the population is not extremely pathological.