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Mathematical Methods
Binomial distribution
Chapter 11
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RESTS ON
Map each elementary outcome to a stated finite random-variable value, combine the masses of outcomes with the same value, and complete or read its probability table
CH 10
Complete and validate a finite probability assignment: every probability is non-negative and the total is exactly one; use stated unequal outcome masses rather than assuming equiprobability
CH 10
Calculate the mean
CH 10
Compute E(X^2) = Σ x^2·Pr(X = x) by squaring each value before multiplying by its probability, and distinguish it from the square of the expected value
CH 10
Extract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical form
CH 2
Form or recognise the union, intersection, difference and complement of finite or interval-defined sets relative to an explicitly stated universe
FOUNDATIONS
Identify a, b, c in zero form and calculate b squared minus 4ac with signs intact
CH 1
Simplify a fraction
CH 1
Evaluate or simplify a bounded factorial expression by expanding only the necessary consecutive factors and using the defining value at the zero index
FOUNDATIONS
Solve a one-step linear equation
CH 7
Signed numbers
CH 3
Interpret unions, intersections and complements as “at least one”, “both”, “only”, “neither” and related event descriptions relative to a stated sample space
FOUNDATIONS
Complete and read a two-event count or probability table, including four interior regions, row and column margins and the grand total
FOUNDATIONS
Solve equations with unknowns on both sides by collecting them onto one side
CH 1
Construct or read a histogram
CH 12
Determine whether two events are independent by checking either conditional invariance or the exact product equality for the intersection
FOUNDATIONS
Bernoulli trials and the Bernoulli distribution
Binomial distribution and parameters
Binomial distribution events
Binomial shape and specified order
Decide whether a described single trial is a Bernoulli trial: exactly two possible outcomes (a success and its complement, the failure), a fixed probability of success, and a single trial
Construct the two-point Bernoulli distribution table from the success probability, and compute E(X) = p, Var(X) = p(1 − p) and SD(X) = √(p(1 − p))
Recover the success probability from a stated Bernoulli variance or standard deviation by solving p(1 − p) = v and selecting the root that satisfies the stated condition (for example p > 1 − p)
State and apply the two defining properties of a discrete probability distribution — every probability is in the interval from zero to one inclusive and the probabilities over all values sum exactly to one — using the properties to validate or reject a distribution
Compute the expected value or mean E(X) = Σ x·Pr(X = x) of a discrete random variable from a table or probability mass function, and interpret it as a long-run average rather than necessarily a value the variable can take
Compute Var(X) = E(X^2) − [E(X)]^2 on exact carriers, and interpret the variance as a spread measure in squared units
Compute SD(X) = √Var(X) as the principal square root of the variance, exactly where the variance is a rational square and at a stated precision otherwise
Distinguish a random experiment, one outcome, the full sample space and an event as a subset of that sample space
Substitute exact coefficients into the quadratic formula, keep the whole denominator 2a, take both sign branches and simplify
Multiply and divide fractions
Rounding
Recognise when a repeated-trial situation is binomial: a fixed number of independent identical trials, two outcomes per trial, and a constant success probability across trials
Construct the full binomial distribution table for every success count from zero to the trial count using Pr(X = x) = C(n, x)(1 − p)^(n − x)p^x, and verify the exact total is one
Compute E(X) = np, Var(X) = np(1 − p) and SD(X) = √(np(1 − p)) for a binomial variable
Recover the success probability and the trial count from a stated binomial mean and variance using 1 − p = Var/mean and then n = mean/p
Calculate the exact probability of a specified number of successes in repeated independent Bernoulli trials using a combination count and complementary success/failure powers; combine disjoint terms for cumulative asks
Compute a combination coefficient exactly from its factorial ratio, cancelling before multiplication and respecting the selection constraints
Solve two-step linear equations by inverse operations in reverse order
Compute cumulative binomial probabilities such as Pr(X ≤ k), Pr(X ≥ k) and at-least-one events by summing exact terms, using the complement where it is shorter
Compute a conditional probability on a binomial variable by identifying the intersection and the conditioning event as success-count sets and dividing the exact intersection sum by the exact conditioning-event sum
Find the smallest integer trial count such that Pr(X ≥ 1) = 1 − (1 − p)^n exceeds a stated threshold, and report the integer count rather than the non-integer solution of the inequality
Obtain the probability of the complement as one minus the event probability and interpret impossible and certain boundary cases
Calculate an exact conditional probability as the intersection probability divided by the conditioning-event probability, with a nonzero conditioning event
Interpret the event after the conditioning event is known and restrict the denominator to that event's outcomes or table margin
Solve a one-variable linear inequation, reversing the order relation exactly when multiplication or division by a negative value occurs
Relate the shape of a binomial probability graph to the success probability: positively skewed when the success probability is below one half, symmetric when it equals one half, negatively skewed when it is above one half, and describe the effect of increasing the trial count
Distinguish a specified success/failure order (multiply the required probabilities in order) from an unordered success count (use the binomial formula with a combination coefficient)
Classify distribution shape
Calculate probabilities of specified success/failure patterns across independent trials, including all, only and at-least-one events, using complements where efficient