1.1Discrete mathematics & applied mathematics

At AP's level 3, building on the mathematical foundations covered in FE, you are expected to quantitatively evaluate the performance of business systems and support decision-making. Whether it is quantifying congestion at a service counter with queueing theory, assessing system reliability with probability and statistics, or finding the optimal allocation of limited resources with linear programming, all of these are practical skills grounded in "reaching a defensible conclusion through calculation." This section emphasizes the procedure and verification pattern behind each calculation.

1.1.1Sets and propositional logic

• A set is a collection of objects. The operations are union (A∪B), intersection (A∩B), and complement (the negation of A). Visualizing with a Venn diagram makes the inclusion-exclusion principle, |A∪B| = |A| + |B| - |A∩B|, for counting elements in two sets intuitive.
• Propositional logic deals with statements (propositions) that are either true or false. For "implies" (A→B), note that it is false only when A is true and B is false. The contrapositive (NOT B -> NOT A) always has the same truth value as the original statement, which is why it is used to reformulate proofs or conditions.

1.1.2Radix numbers and error

• An n-ary (radix-n) number represents values in any base n, such as binary, octal, or hexadecimal. As an application of base conversion, multiplying each digit by its positional weight n^k and summing gives a unified method for converting between any two bases.
• Floating-point error (rounding error, cancellation, loss of trailing digits) is shared ground with FE, but AP also tests practical mitigation decisions—such as managing money as integers/fixed point, or reordering a calculation to avoid cancellation.

1.1.3Probability & statistics (expectation, variance, normal distribution, correlation/regression)

• Expected value is the sum of each outcome multiplied by its probability (the average value you can expect): E(X) = sum of x_i * p_i. Variance measures how spread out the values are from the expected value, computed as V(X) = E(X^2) - {E(X)}^2 (the expected value of the square minus the square of the expected value).
• The normal distribution is a symmetric, bell-shaped continuous distribution centered on the mean. It is used to model quality control data and test-score distributions, and the empirical rule that about 68% of data falls within ±1 standard deviation and about 95% within ±2 standard deviations is a frequently tested fact.
• The correlation coefficient measures the strength of a linear relationship between two variables on a scale of -1 to 1. Values near 1 indicate a strong positive correlation, near -1 a strong negative correlation, and near 0 little to no correlation. Regression analysis models the relationship between correlated variables as an equation (a regression line) to predict one from the other.

1.1.4Queueing theory (M/M/1)

• Queueing theory mathematically models congestion in a service point or system where arrivals and processing occur randomly. The M/M/1 model is the most basic queueing model: arrivals and service times are both exponentially distributed (random), with a single server.
• Utilization ρ (rho) is the fraction of time the server is busy: ρ = λ / μ (λ = mean arrival rate, μ = mean service rate). Unless ρ < 1, the queue grows without bound and the system breaks down.
• In the M/M/1 model, the average waiting time (time spent in the queue) is Wq = ρ / (μ - λ). The average time in system (queueing time plus service time) is W = 1 / (μ - λ). Increasing μ (more servers, faster processing) or decreasing λ is the basic way to lower ρ.

Suppose a call center receives an average of 12 inquiries per hour (λ=12/hr) and one operator can handle an average of 15 per hour (μ=15/hr). First, the utilization ρ is ρ = λ / μ = 12 / 15 = 0.8 (80%), and since ρ < 1, the queue is stable. Next, the average waiting time is Wq = ρ / (μ - λ) = 0.8 / (15 - 12) = 0.8 / 3 ≈ 0.267 hours, which converts to 0.267 × 60 ≈ 16 minutes. The average time in system (waiting plus service) is W = 1 / (μ - λ) = 1 / 3 hour ≈ 20 minutes, and as a sanity check, W - Wq = 20 - 16 = 4 minutes matches the average service time (1/μ = 1/15 hour = 4 minutes). Suppose management now wants to halve the average wait to 8 minutes or less: solving Wq' ≤ 8/60 hour with ρ'=12/μ' backward gives μ' ≥ 18, translating directly into the concrete measure of raising operator throughput from 15/hr to 18/hr (via more staff or process improvement). Queueing theory turns a "felt sense of congestion" into a quantified basis for decisions.

1.1.5Optimization (linear programming, shortest path)

• Linear programming finds the variable values that maximize or minimize an objective function (a linear expression such as profit or cost) subject to multiple constraints (linear inequalities). With two variables, the optimum occurs at a vertex of the feasible region formed by the constraints (the vertex-enumeration method).
• The shortest-path problem finds the path between two nodes in a graph that minimizes the total cost (distance, time, etc.). Dijkstra's algorithm is the representative solution method, which fixes the shortest distance to nodes in order of increasing distance from the source.

1.1.6Section summary

• Queueing (M/M/1): ρ=λ/μ, Wq=ρ/(μ-λ), W=1/(μ-λ). Match the units of λ and μ, and assume ρ<1
• Variance = E(X^2) - {E(X)}^2; the normal distribution has about 68% of data within ±1σ and about 95% within ±2σ
• Linear programming searches the vertices of the feasible region for the optimum, and the shortest path is found via methods such as Dijkstra's algorithm