School / IB / MATH AA HL / Statistics and probability Question practice
Statistics and probability About this practice Extends SL probability with Bayes' theorem, the general variance formula for a discrete random variable, and continuous random variables.
Mathematics AA: Higher Level Statistics and probability
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1 X ~ B(10, 0.3). Find P(X = 3). Paper 2 style Medium 5 marks Calculator + 2 Marking analysis: A learner attempts the following task: “X ~ B(10, 0.3). Find P(X = 3).” Their response addresses only this point: “States the binomial probability formula P(X = k) = ⁿCₖ pᵏ(1 − p)ⁿ⁻ᵏ.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 3 X ~ B(20, 0.25). Find E(X) and Var(X). Paper 2 style Medium 4 marks Calculator + 4 Marking analysis: A learner attempts the following task: “X ~ B(20, 0.25). Find E(X) and Var(X).” Their response addresses only this point: “States the formulas E(X) = np and Var(X) = np(1 − p).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 5 A discrete random variable X has the probability distribution P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3. Find E(X) and Var(X). Paper 2 style Hard 6 marks Calculator + 6 Marking analysis: A learner attempts the following task: “A discrete random variable X has the probability distribution P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3. Find E(X) and Var(X).” Their response addresses only this point: “States E(X) = Σx·P(X = x).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks Calculator + 7 A continuous random variable X has probability density function f(x) = kx for 0 ≤ x ≤ 4, and f(x) = 0 otherwise. Find the value of k, and hence find P(X > 2). Paper 2 style Hard 6 marks Calculator + 8 Marking analysis: A learner attempts the following task: “A continuous random variable X has probability density function f(x) = kx for 0 ≤ x ≤ 4, and f(x) = 0 otherwise. Find the value of k, and hence find P(X > 2).” Their response addresses only this point: “States that a valid pdf must satisfy ∫₀⁴ f(x) dx = 1.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks Calculator + 9 A continuous random variable X has probability density function f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Find E(X). Paper 1 style Medium 4 marks No calculator + 10 Marking analysis: A learner attempts the following task: “A continuous random variable X has probability density function f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Find E(X).” Their response addresses only this point: “States the formula for the expectation of a continuous random variable: E(X) = ∫x f(x) dx over the domain.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 11 A rare disease affects 1% of a population. A test for the disease is 95% accurate for people who have it (true positive) and gives a false positive for 5% of people who do not have it. A person tests positive. Find the probability that they actually have the disease. Paper 2 style Hard 6 marks Calculator + 12 Marking analysis: A learner attempts the following task: “A rare disease affects 1% of a population. A test for the disease is 95% accurate for people who have it (true positive) and gives a false positive for 5% of people who do not have it. A person tests positive. Find the probability that they actually have the disease.” Their response addresses only this point: “States Bayes' theorem: P(D | +) = P(+ | D)P(D) / P(+).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks Calculator + 13 Factory A supplies 60% of a component and Factory B supplies the remaining 40%. 2% of Factory A's components are defective, and 5% of Factory B's components are defective. A component is chosen at random and found to be defective. Find the probability it came from Factory B. Paper 2 style Hard 6 marks Calculator + 14 Marking analysis: A learner attempts the following task: “Factory A supplies 60% of a component and Factory B supplies the remaining 40%. 2% of Factory A's components are defective, and 5% of Factory B's components are defective. A component is chosen at random and found to be defective. Find the probability it came from Factory B.” Their response addresses only this point: “States Bayes' theorem: P(B | D) = P(D | B)P(B) / P(D).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks Calculator + 15 A continuous random variable X has probability density function f(x) = x/8 for 0 ≤ x ≤ 4, and f(x) = 0 otherwise. Find the median of X. Paper 2 style Medium 5 marks Calculator + 16 Marking analysis: A learner attempts the following task: “A continuous random variable X has probability density function f(x) = x/8 for 0 ≤ x ≤ 4, and f(x) = 0 otherwise. Find the median of X.” Their response addresses only this point: “States that the median m satisfies ∫₀ᵐ f(x) dx = 0.5.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 17 The heights of adult males are normally distributed with mean 175 cm and standard deviation 7 cm. Find the probability that a randomly selected male is taller than 185 cm. Paper 2 style Medium 4 marks Calculator + 18 Marking analysis: A learner attempts the following task: “The heights of adult males are normally distributed with mean 175 cm and standard deviation 7 cm. Find the probability that a randomly selected male is taller than 185 cm.” Their response addresses only this point: “States X ~ N(175, 7²).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 19 The number of emails a person receives per hour follows a Poisson distribution with mean 4. Find the probability that exactly 6 emails are received in a given hour. Paper 2 style Medium 4 marks Calculator + 20 Marking analysis: A learner attempts the following task: “The number of emails a person receives per hour follows a Poisson distribution with mean 4. Find the probability that exactly 6 emails are received in a given hour.” Their response addresses only this point: “States X ~ Po(4).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 21 A continuous random variable X has probability density function f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Given that E(X) = 3/4, find Var(X). Paper 2 style Medium 5 marks Calculator + 22 Marking analysis: A learner attempts the following task: “A continuous random variable X has probability density function f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Given that E(X) = 3/4, find Var(X).” Their response addresses only this point: “States Var(X) = E(X²) − [E(X)]².” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 23 For two events A and B, P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. Find P(A | B). Paper 2 style Easy 3 marks Calculator + 24 Marking analysis: A learner attempts the following task: “For two events A and B, P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. Find P(A | B).” Their response addresses only this point: “States P(A | B) = P(A ∩ B)/P(B).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 25 X and Y are independent random variables with E(X) = 5, E(Y) = 3, Var(X) = 4 and Var(Y) = 9. Find E(X + Y) and Var(X + Y). Paper 2 style Easy 3 marks Calculator + 26 Marking analysis: A learner attempts the following task: “X and Y are independent random variables with E(X) = 5, E(Y) = 3, Var(X) = 4 and Var(Y) = 9. Find E(X + Y) and Var(X + Y).” Their response addresses only this point: “States that E(X + Y) = E(X) + E(Y), which holds regardless of independence.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 27 X is a random variable with E(X) = 10 and Var(X) = 4. Find E(3X − 2) and Var(3X − 2). Paper 2 style Easy 3 marks Calculator + 28 Marking analysis: A learner attempts the following task: “X is a random variable with E(X) = 10 and Var(X) = 4. Find E(3X − 2) and Var(3X − 2).” Their response addresses only this point: “States E(aX + b) = aE(X) + b.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 29 A hand of 5 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 aces. Paper 2 style Medium 5 marks Calculator + 30 Marking analysis: A learner attempts the following task: “A hand of 5 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 aces.” Their response addresses only this point: “States the total number of possible 5-card hands: ⁵²C₅.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + Self-assessed Not marked yet
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