333 questions · 35 question types identified
Questions providing a partial probability distribution with two unknown constants and asking to find them using the constraint that probabilities sum to 1 and a given expectation value.
A question is this type if and only if it describes a random experiment or game and asks to construct the complete probability distribution table for a defined random variable.
Probabilities given as a single algebraic expression in x (e.g., kx, kx², k(x²-1), k(x+1)) where k is found by summing over all values of x and setting equal to 1.
Questions providing a partial probability distribution with one unknown constant (or multiple unknowns with a simple relationship) and asking to find it using only the constraint that probabilities sum to 1.
The probability distribution is explicitly provided in a table or formula, and the question asks to calculate E(X) directly using the standard formula.
Questions that provide a complete probability distribution table and ask to calculate Var(X), possibly also asking for E(X) first.
| \(r\) | 10 | 20 | 30 | 40 |
| \(\mathrm { P } ( X = r )\) | 0.2 | 0.3 | 0.3 | 0.2 |
Questions providing a partial probability distribution with multiple unknown constants and asking to find them using constraints including both a given expectation and a given variance.
| \(x\) | - 3 | 0 | 2 | 4 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(q\) | \(r\) | 0.4 |
A question is this type if and only if the probability function is given as a piecewise or conditional formula and asks to construct the distribution table or find properties.
Questions where the probability is calculated from independent events such as coin tosses, dice rolls, or spinner spins using multiplication of independent probabilities.
| \(r\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 1 } { 216 }\) | \(\frac { 7 } { 216 }\) | \(\frac { 19 } { 216 }\) | \(\frac { 37 } { 216 }\) | \(\frac { 61 } { 216 }\) | \(\frac { 91 } { 216 }\) |
A question is this type if and only if it describes a process that continues until a success occurs or a maximum number of trials is reached, and asks for the distribution of the number of trials.
Questions where the probability is calculated from selecting items without replacement from a finite collection, requiring combinations or systematic enumeration of outcomes.
| \(r\) | 10 | 15 | 20 | 25 | 30 |
| \(\mathrm { P } ( X = r )\) | 0.1 | 0.4 | 0.1 | 0.2 | 0.2 |
Questions where a complete or partial probability distribution is explicitly given in a table or formula, and the task is to calculate probabilities directly using addition or the complement rule.
A question is this type if and only if it involves selecting items without replacement from a finite population and asks for the distribution of the number of items with a certain property.
A question is this type if and only if it involves a game, lottery, or business scenario and asks to calculate expected profit, cost, or revenue.
Probabilities defined by different expressions for different ranges of x (e.g., kx for some values and k(x+1) for others, or k(1-x)² for some values and 0 otherwise), requiring separate treatment of each piece when finding k.
A question is this type if and only if it asks to find or use the cumulative distribution function F(x) = P(X ≤ x).
Probabilities already displayed in a table with expressions involving k (e.g., 3c, 4c, 5c or 4p, 5p², 1.5p) where k is found by summing the table entries and setting equal to 1.
| \(x\) | 1 | 4 | 5 | 7 | 9 |
| \(\mathrm { P } ( X = x )\) | \(4 p\) | \(5 p ^ { 2 }\) | \(1.5 p\) | \(2.5 p\) | \(1.5 p\) |
Questions where the probability is calculated by substituting a specific value into a given probability distribution formula to verify it produces the stated result.
Questions asking for the probability distribution or properties (mean, variance) of the sum, product, or other arithmetic combination of exactly two independent values from the same distribution.
| \(y\) | 1 | 2 | 3 |
| \(\mathrm { P } ( Y = y )\) | 0.2 | 0.3 | 0.5 |
| \(z\) | 1 | 2 | 3 |
| \(\mathrm { P } ( Z = z )\) | 0.1 | 0.25 | 0.65 |
Questions that give E(X) and/or Var(X) directly and ask to apply the standard formulas E(aX+b)=aE(X)+b or Var(aX+b)=a²Var(X) without needing to calculate from a distribution.
Questions that ask specifically for the variance of a linear transformation Var(aX+b) after finding or using Var(X), requiring the variance transformation formula.
Questions where the random variable is defined as the sum or difference of the outcomes from two spinners or dice.
A question is this type if and only if it asks to identify the mode (most likely value) of a discrete random variable.
The question requires first constructing or deriving the probability distribution from a scenario (dice, games, matching problems) before calculating E(X).
Questions asking about the relationship between two independent values, such as which is larger, whether they differ, or conditional probabilities based on their relative ordering.
| \(x\) | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.6 | 0.3 | 0.1 |
Questions involving three or more independent values from the same distribution, asking about their sum, product, or other combined properties.
| \(x\) | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | 0.1 | 0.3 | 0.4 | 0.2 |
Questions that ask to derive expressions for E(aX+b) or Var(aX+b) in terms of given parameters, or to identify which formula is correct from multiple choices.
Questions where the probability distribution must first be constructed from a scenario (such as tree diagrams, combinatorial setups, or deriving probabilities from constraints) before calculating the required probability.
| \(r\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 1 } { 216 }\) | \(\frac { 7 } { 216 }\) | \(\frac { 19 } { 216 }\) | \(\frac { 37 } { 216 }\) | \(\frac { 61 } { 216 }\) | \(\frac { 91 } { 216 }\) |
Questions that provide a probability function formula (e.g., P(X=x) = kx²) and require finding the constant, then calculating Var(X) or Var(aX+b).
A question is this type if and only if it asks to find P(A|B) where A and B are events defined in terms of a discrete random variable.
The cumulative distribution function F(X) is given, requiring conversion to probability distribution P(X=x) before calculating E(X).
| VIAV SIHI NI III IM IONOOC | VIIIV SIHI NI III IM I I N O O | VI4V SIHI NI III IM I ON OC |
A question is this type if and only if it defines one random variable Y in terms of another X (e.g., Y = aX + b) and asks to find properties of Y from properties of X.
Questions where the random variable is defined as the product of the outcomes from two spinners or dice.
Questions where the random variable is the minimum of two outcomes, or involves a conditional rule (e.g., smaller if different, zero if equal).
Questions that provide a complete probability distribution (table or function) and ask to find E(aX+b) or Var(aX+b), requiring calculation of E(X) and/or Var(X) first.