203 questions · 25 question types identified
Questions that directly give n and p and ask for P(X = k), P(X ≤ k), or P(X ≥ k) with no additional context or multi-part scenarios.
Find P(a ≤ X ≤ b) or P(a < X < b) for a binomial random variable, requiring subtraction of cumulative probabilities.
Questions that ask students to state or suggest a suitable binomial distribution model including specific parameter values for a given context.
Determine n or p given information about mean, variance, or specific probabilities like P(X = 0) or mode.
Calculate expected number of occurrences across multiple independent repetitions of a single binomial experiment (e.g., expected number of samples with a specific outcome across many samples).
Questions that ask students to state assumptions or verify conditions for a specific given scenario to be modeled by a binomial distribution.
Calculate the probability that a certain number of independent binomial samples satisfy a given condition (e.g., exactly 2 out of 3 periods have at least one success).
Questions involving several people/groups each with their own binomial distribution, or multiple separate binomial probability calculations in different parts (e.g., boys and girls with different probabilities, or different days with different probabilities).
Solve problems involving binomial distribution combined with other probability concepts like conditional probability or tree diagrams.
Questions that ask directly for the mean and/or variance of a binomial distribution X ~ B(n,p), or simple linear transformations Y = aX + b, without requiring probability calculations.
Find the probability that the first success occurs on a specific trial, or before/after a certain trial.
Questions that ask students to state or list the general conditions required for a binomial distribution without reference to a specific context.
Finding minimum n such that P(X ≥ 1) exceeds a threshold, typically solved using P(X ≥ 1) = 1 - P(X = 0) = 1 - (1-p)^n.
Finding minimum or maximum n for other probability conditions (e.g., P(X < r) < k, P(X = 0) ≥ k, or mean requirements), requiring cumulative binomial calculations or inequalities.
Use normal distribution with continuity correction to approximate binomial probabilities for large n.
Complete or construct the full probability distribution table for a binomial or related discrete random variable.
Find probabilities involving the sum or combined outcomes of multiple independent values drawn from the same binomial distribution (e.g., sum of two values equals a target).
Questions that require calculating both probabilities (using binomial formula or tables) and expected value/variance in the same problem, typically in multi-part questions.
Calculate expected count where the event of interest is defined by a binomial probability within each batch, then count across multiple batches (e.g., expected number of boxes containing exactly k items with a property).
Problems where a binomial outcome determines parameters for a second binomial experiment.
Find the value of X with highest probability, often the value closest to the mean.
Questions where n or p must be derived from given information (e.g., 'average number of 5s was 4.8' to find p, or 'mean number of defectives is 1.6' in packs of 20) before calculating probabilities.
Questions requiring formal proof from first principles of the binomial mean and variance formulas, or deriving variance using E(X(X-1)).
Questions where the binomial setup involves calculating probabilities for complementary scenarios or 'at least one' type problems that require understanding of complement rule in the context setup.
Find probabilities or distributions involving sums, products, or other combinations of independent binomial random variables.