Mathematics AA: Higher Level
Statistics and probability — Topic 4
- 1.
X ~ B(10, 0.3). Find P(X = 3).
[5 marks] - 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.
[5 marks] - 3.
X ~ B(20, 0.25). Find E(X) and Var(X).
[4 marks] - 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.
[4 marks] - 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).
[6 marks] - 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.
[6 marks] - 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).
[6 marks] - 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.
[6 marks] - 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).
[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.
[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.
[6 marks] - 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.
[6 marks] - 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.
[6 marks] - 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.
[6 marks] - 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.
[5 marks] - 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.
[5 marks] - 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.
[4 marks] - 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.
[4 marks] - 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.
[4 marks] - 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.
[4 marks] - 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).
[5 marks] - 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.
[5 marks] - 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).
[3 marks] - 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.
[3 marks] - 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).
[3 marks] - 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.
[3 marks] - 27.
X is a random variable with E(X) = 10 and Var(X) = 4. Find E(3X − 2) and Var(3X − 2).
[3 marks] - 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.
[3 marks] - 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.
[5 marks] - 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.
[5 marks]