AP Exams (Advanced Placement) AP Mathematics and Computer Science (Statistics & CS) Syllabus

State the complement rule and the general addition rule for probability.

Complement: P(Aᶜ) = 1 − P(A). Addition: P(A or B) = P(A) + P(B) − P(A and B).

What does it mean for two events to be mutually exclusive (disjoint), and what does that imply for P(A and B)?

They cannot occur at the same time; P(A and B) = 0, so P(A or B) = P(A) + P(B).

State the formula for conditional probability and the test for independence.

P(A | B) = P(A and B) / P(B). Events A and B are independent if P(A | B) = P(A), equivalently P(A and B) = P(A)·P(B).

How do you find the mean (expected value) of a discrete random variable X?

E(X) = μ = Σ [xᵢ · P(xᵢ)] — sum each value times its probability.

What four conditions define a binomial setting (BINS)?

Binary outcomes (success/failure), Independent trials, fixed Number of trials n, and Same probability p of success on each trial.

State the mean and standard deviation of a binomial random variable.

μ = np and σ = sqrt(np(1−p)), where n is the number of trials and p is the probability of success.

For independent random variables X and Y, how do you combine their means and variances?

Means add: μ(X±Y) = μX ± μY. Variances ADD for both sum and difference: σ²(X±Y) = σ²X + σ²Y (standard deviations do NOT add directly).

What does the 68–95–99.7 (empirical) rule state for a Normal distribution?

About 68% of data lie within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs.

What does the Central Limit Theorem state about the sampling distribution of the sample mean?

For a large enough sample size (n ≥ 30 as a rule of thumb), the sampling distribution of x̄ is approximately Normal regardless of the population's shape.

Write the general form of a confidence interval.

point estimate ± (critical value)(standard error of the estimate).

Interpret what '95% confidence' means in a confidence interval.

If we repeated the sampling process many times, about 95% of the resulting intervals would capture the true population parameter (the method succeeds 95% of the time).

State the confidence interval formula for a single population proportion.

p̂ ± z* · sqrt( p̂(1−p̂)/n ), where z* is the critical value from the standard Normal distribution.

How does increasing the sample size affect the margin of error, all else equal?

It decreases the margin of error (margin of error is proportional to 1/√n), making the interval narrower.

What is the difference between a null hypothesis (H₀) and an alternative hypothesis (Hₐ)?

H₀ states no effect/no difference (the status quo, with an equals sign); Hₐ states the effect/difference being tested (uses <, >, or ≠).

Define the P-value in hypothesis testing.

The probability, assuming H₀ is true, of obtaining a test statistic at least as extreme as the one observed. A small P-value is evidence against H₀.

State the decision rule comparing a P-value to the significance level α.

If P-value ≤ α, reject H₀ (statistically significant); if P-value > α, fail to reject H₀.

Distinguish between a Type I error and a Type II error.

Type I: rejecting H₀ when it is actually true (false positive), with probability α. Type II: failing to reject H₀ when it is actually false (false negative), with probability β.

What is the power of a significance test, and how is it related to β?

Power is the probability of correctly rejecting a false H₀; Power = 1 − β. Larger samples and larger effect sizes increase power.

What does a chi-square goodness-of-fit test assess?

Whether the observed distribution of one categorical variable matches a claimed/expected distribution. χ² = Σ (observed − expected)² / expected.

Differentiate the chi-square test for homogeneity from the chi-square test for independence/association.

Homogeneity compares the distribution of a categorical variable across two or more populations/groups; independence tests whether two categorical variables are associated within a single population. Both use the same χ² statistic and (rows−1)(cols−1) degrees of freedom.

In inference for the slope of a regression line, what are the hypotheses and test statistic?

H₀: β = 0 (no linear relationship) vs Hₐ: β ≠ 0 (or </>). Test statistic t = b / SE(b) with df = n − 2, using a t-distribution.

In Java, what are the eight primitive data types?

byte, short, int, long, float, double, char, and boolean.

In Java (AP CS A), what is the difference between == and the .equals() method for objects like String?

== compares references (whether two variables point to the same object); .equals() compares the contents/values of the objects.

What is the difference between a while loop and a for loop in Java?

A while loop repeats while a condition is true and is best when the number of iterations is unknown; a for loop bundles initialization, condition, and update and is best when the count of iterations is known.

In Java, what is the difference between a class and an object?

A class is a blueprint/template defining attributes (instance variables) and behaviors (methods); an object is a specific instance of a class created with the new keyword (a constructor call).

What is the purpose of a constructor in Java, and what is encapsulation?

A constructor initializes a new object's instance variables when it is created. Encapsulation is hiding internal data using private fields and exposing access through public methods (getters/setters).

What is inheritance in Java, and what keyword establishes it?

Inheritance lets a subclass acquire the fields and methods of a superclass, enabling code reuse; the keyword 'extends' establishes the relationship. super() calls the superclass constructor.

How does a Java array differ from an ArrayList?

An array has a fixed size and can hold primitives or objects, accessed with [ ]; an ArrayList is resizable, holds objects (or wrappers) only, and uses methods like add, get, set, size, and remove.

In AP CS A, what is the difference between traversing a 1D array and a 2D array?

A 1D array is traversed with one loop over indices 0..length−1; a 2D array (array of arrays) is traversed with nested loops over rows (arr.length) and columns (arr[row].length).

What two essential components must every recursive method have?

A base case that stops the recursion (returns without recursing) and a recursive case that calls the method on a smaller/simpler input moving toward the base case.

Compare linear search and binary search, including their requirements.

Linear search checks each element in order (works on any list, O(n)); binary search repeatedly halves a SORTED list (O(log n)), requiring the data to be sorted first.

In AP CSP, what is an algorithm, and what three building blocks can express any algorithm?

An algorithm is a finite, step-by-step set of instructions to accomplish a task. The three control structures are sequencing, selection (if/else), and iteration (loops).

In AP CSP, distinguish a parameter from an argument, and explain a procedure's benefit of abstraction.

A parameter is the variable named in a procedure's definition; an argument is the actual value passed in when calling it. Procedures with parameters provide abstraction by hiding implementation detail and enabling reuse.

In AP CSP Creative Development, why is collaboration and iterative development valued in program design?

Collaboration brings diverse perspectives that reduce bias and errors; an iterative process (investigate, design, prototype, test, refine) catches problems early and steadily improves the program.

In AP CSP, how is analog data different from digital data, and why must analog be sampled?

Analog data is continuous (e.g., sound waves); digital data is discrete, stored as bits. Analog signals must be sampled (measured at intervals) to be represented digitally; higher sampling rates give better fidelity.

In AP CSP, distinguish lossy from lossless data compression.

Lossless compression reduces size with no information lost (data fully restorable, e.g., PNG/ZIP); lossy compression discards some data for greater size reduction and cannot be perfectly restored (e.g., JPEG, MP3).

In AP CSP, what is the difference between a problem solvable by an algorithm and an undecidable problem, and what is a heuristic?

Some problems have no algorithm that always gives a correct answer (undecidable). A heuristic is an approach that produces a good-enough (not guaranteed optimal) solution quickly, often for problems too slow to solve exactly (e.g., unreasonable-time problems).

In AP CSP, contrast a problem that runs in reasonable time with one that runs in unreasonable time.

Reasonable time: the number of steps grows polynomially with input size (e.g., constant, linear, square). Unreasonable time: steps grow faster, like exponentially or factorially, becoming impractical for large inputs.

In AP CSP, what is the purpose of redundancy and routing in the Internet, and what is fault tolerance?

Redundancy provides multiple paths between points; routing dynamically chooses paths so data (packets) still arrive if some paths fail. This redundancy gives the network fault tolerance — it keeps working despite component failures.

In AP CSP, what is the difference between bandwidth and latency, and what role do protocols like TCP/IP play?

Bandwidth is the maximum data transmitted per unit time; latency is the delay before data transfer begins. Protocols (TCP/IP, HTTP, DNS) are agreed-upon rules that let different devices communicate reliably across networks.

In AP CSP, explain the difference between parallel computing and distributed computing.

Parallel computing uses multiple processors to carry out parts of a task simultaneously on one machine; distributed computing uses multiple networked computers working together on a shared task.

In AP CSP Impact of Computing, what is the digital divide and what is meant by computing bias?

The digital divide is the unequal access to computing technology/the Internet across groups or regions. Computing bias is when programs (often via their data or design) systematically favor or disadvantage certain groups, reflecting human bias.

In AP CSP, define personally identifiable information (PII) and name two safeguards for protecting data.

PII is information that can identify an individual (name, SSN, address, location). Safeguards include strong passwords/multifactor authentication and encryption (and limiting what data is shared).

In AP CSP, what is the difference between symmetric and public key (asymmetric) encryption?

Symmetric encryption uses the same secret key to encrypt and decrypt; public key encryption uses a public key to encrypt and a separate private key to decrypt, removing the need to share a secret key.

In AP CSP Impact of Computing, what is the distinction between open-source and creative ownership concerns like intellectual property?

Open-source software is freely shared for use and modification; intellectual property (protected by copyright/licenses) restricts unauthorized copying. Creative Commons licenses let creators specify how their work may be reused.