Unit 4 — Random Variables and Probability Distributions · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 82 questions on this lesson, each with an explanation.
If X ~ B(12, 0.3), then P(X = 5) = ……
Correct answer: (C) ¹²C₅ × (0.3)⁵ × (0.7)⁷
Explanation: Step 1: P(X = r) = ⁿCᵣ × Sʳ × (1 − S)ⁿ⁻ʳ with n = 12, r = 5 and S = 0.3. Step 2: P(X = 5) = ¹²C₅ × (0.3)⁵ × (1 − 0.3)¹²⁻⁵ = ¹²C₅ × (0.3)⁵ × (0.7)⁷ ∴ P(X = 5) = ¹²C₅ × (0.3)⁵ × (0.7)⁷ (leaving out ¹²C₅ ignores the number of ways the 5 successes can be placed among the 12 trials).
If X ~ B(20, 0.3), then the expectation μ = ……
Correct answer: (D) 6
Explanation: Step 1: μ = n × S Step 2: μ = 20 × 0.3 = 6 ∴ μ = 6 (4.2 = n × S × (1 − S) is the variance, and 14 = n × (1 − S) is the expected number of failures).
If X ~ G(0.4), then P(X = 0) = ……
Correct answer: (A) zero
Explanation: Step 1: In P(X = n) = S(1 − S)ⁿ⁻¹, n = 1, 2, 3, … because the first success needs at least one trial. Step 2: So X = 0 is impossible. ∴ P(X = 0) = zero.
If X ~ B(5, 0.4), then P(X = 3) = ……
Correct answer: (A) 0.2304
Explanation: Step 1: P(X = r) = ⁿCᵣ × Sʳ × (1 − S)ⁿ⁻ʳ with n = 5, r = 3 and S = 0.4. Step 2: P(X = 3) = ⁵C₃ × (0.4)³ × (0.6)² = 10 × 0.064 × 0.36 = 0.2304 ∴ P(X = 3) = 0.2304 (0.3456 = ⁵C₃ × (0.6)³ × (0.4)² swaps S and 1 − S, and 0.0230 forgets ⁵C₃).
If X ~ B(20, 0.4) and Y ~ B(40, 0.2), which of the following statements is true when we move from X to Y?
Correct answer: (A) The expectation stays 8 and the variance increases from 4.8 to 6.4
Explanation: Step 1: Expectation of X = 20 × 0.4 = 8 and expectation of Y = 40 × 0.2 = 8. Step 2: Variance of X = 20 × 0.4 × 0.6 = 4.8 and variance of Y = 40 × 0.2 × 0.8 = 6.4. ∴ The expectation stays 8 and the variance increases from 4.8 to 6.4 (doubling n and halving S keeps n × S the same, but 1 − S changes).
A regular coin is tossed 30 times. The probability that a head appears exactly 10 times is ……
Correct answer: (C) ³⁰C₁₀ × (1/2)³⁰
Explanation: Step 1: Let X be the number of heads, X ~ B(30, 1/2). Step 2: P(X = 10) = ³⁰C₁₀ × (1/2)¹⁰ × (1/2)²⁰ = ³⁰C₁₀ × (1/2)³⁰ ∴ The probability is ³⁰C₁₀ × (1/2)³⁰ (³⁰C₁₀ × (1/2)¹⁰ forgets the probability of failure in the other 20 tosses, and ¹⁰C₃₀ has n and r the wrong way round).
A regular die is rolled repeatedly until a number greater than 4 appears for the first time. The variance of the number of rolls needed is ……
Correct answer: (A) 6
Explanation: Step 1: The numbers greater than 4 are 5 and 6, so S = 2/6 = 1/3 and X ~ G(1/3). Step 2: σ² = (1 − S)/S² = (2/3) ÷ (1/9) = 6 ∴ The variance is 6 (3 = 1/S is the expectation, 2 = (1 − S)/S forgets the square, and √6 is the standard deviation).
If X is a binomial random variable with expectation μ = 3 and variance σ² = 1.2, then P(X ≥ 4) = ……
Correct answer: (D) 0.33696
Explanation: Step 1: 1 − S = σ² ÷ μ = 1.2 ÷ 3 = 0.4, so S = 0.6 and n = μ ÷ S = 3 ÷ 0.6 = 5. Step 2: P(X ≥ 4) = ⁵C₄ × (0.6)⁴ × (0.4)¹ + ⁵C₅ × (0.6)⁵ = 0.2592 + 0.07776 = 0.33696 ∴ P(X ≥ 4) = 0.33696 (0.2592 is P(X = 4) alone, and 0.08704 comes from swapping S and 1 − S).
If X ~ G(S) and P(X > 2) = 0.49, then P(X = 3) = ……
Correct answer: (B) 0.147
Explanation: Step 1: P(X > 2) = (1 − S)² = 0.49, so 1 − S = 0.7 and S = 0.3. Step 2: P(X = 3) = 0.3 × (0.7)² = 0.147 ∴ P(X = 3) = 0.147 (0.063 = 0.7 × (0.3)² takes S = 0.7 by mistake, and 0.343 = (0.7)³ is P(X > 3)).
Ahmed tosses a regular coin 3 times and Omar tosses another regular coin twice, all tosses being independent. The probability that they get the same number of heads is ……
Correct answer: (C) 5/16
Explanation: Step 1: For Ahmed, X ~ B(3, 1/2): P(X = 0) = 1/8, P(X = 1) = 3/8, P(X = 2) = 3/8, P(X = 3) = 1/8. For Omar, Y ~ B(2, 1/2): P(Y = 0) = 1/4, P(Y = 1) = 1/2, P(Y = 2) = 1/4. Step 2: P(X = Y) = (1/8)(1/4) + (3/8)(1/2) + (3/8)(1/4) = 1/32 + 6/32 + 3/32 = 10/32 = 5/16 ∴ The probability is 5/16 (3/16 takes only the case of one head each, and 1/2 is the probability that Ahmed gets more heads).
The first 10 questions in every subject are free, each with an explanation.