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 77 questions on this lesson, each with an explanation.
If f is the probability distribution function of a discrete random variable X, which of the following cannot be a value of f(xᵣ)?
Correct answer: (C) 1.2
Explanation: Step 1: The two conditions: f(xᵣ) ≥ 0 for each r, and f(x₁) + f(x₂) + … + f(xₙ) = 1. Step 2: So no value of f(xᵣ) can be greater than 1, while 0 and 1 are both allowed. ∴ 1.2 cannot be a value of f(xᵣ) (it would make the sum greater than 1).
The random variable is a function whose domain is …… and whose co-domain is the set of real numbers R.
Correct answer: (D) the set of the elements of the sample space S
Explanation: Step 1: By definition, the random variable is a function whose domain is the set of the elements of the sample space S and whose co-domain is the set of real numbers R, i.e. X: S → R. ∴ the domain is the sample space S. (The interval [0, 1] is where probability values lie; it is not the domain of X.)
If X is a discrete random variable whose range is {0, 1, 2, 3} and P(X = 1) = 0.25, P(X = 2) = 0.4, P(X = 3) = 0.15, then P(X = 0) = ……
Correct answer: (C) 0.2
Explanation: Step 1: The sum of the probabilities = 1. Step 2: P(X = 0) = 1 − (0.25 + 0.4 + 0.15) = 1 − 0.8 ∴ P(X = 0) = 0.2 (0.8 is a trap: it is the sum of the given probabilities).
For a spinner, the random variable X expresses the number at which the pointer stops, and its probability distribution is xᵣ: 1, 2, 3 and f(xᵣ): 0.2, 0.5, 0.3. If the spinner is spun twice independently, then the probability that the pointer stops at the same number both times = ……
Correct answer: (D) 0.38
Explanation: Step 1: The event is made of three mutually exclusive events: (1, 1) or (2, 2) or (3, 3), and each probability is a product because the spins are independent. Step 2: 0.2² + 0.5² + 0.3² = 0.04 + 0.25 + 0.09 ∴ the probability = 0.38 (0.62 is the probability of two different numbers, and 0.25 is the probability of (2, 2) only).
A bag contains 3 red balls and 2 white balls. Two balls are drawn one after another without replacement. If X expresses the number of red balls drawn, then P(X = 1) = ……
Correct answer: (B) 3/5
Explanation: Step 1: Without replacement, n(S) = 5 × 4 = 20. Step 2: X = 1 means red then white: 3 × 2 = 6, or white then red: 2 × 3 = 6, a total of 12 outcomes. ∴ P(X = 1) = 12/20 = 3/5 (3/10 counts one order only, and 12/25 draws with replacement).
A box contains 3 cards numbered 1, 2, 3. Two cards are drawn one after another with replacement. If X expresses the product of the two numbers on the drawn cards, then P(X ≥ 3) = ……
Correct answer: (D) 2/3
Explanation: Step 1: With replacement, n(S) = 3 × 3 = 9. Step 2: The product is ≥ 3 in (1, 3), (3, 1), (2, 2), (2, 3), (3, 2), (3, 3), that is 6 outcomes. ∴ P(X ≥ 3) = 6/9 = 2/3 (4/9 is a trap: it leaves out the product 3, so it is P(X > 3)).
A fair coin and a regular die are thrown together. The random variable X is the number on the die plus 1 if a Head appears, or the number on the die as it is if a Tail appears. Then P(X ≥ 6) = ……
Correct answer: (B) 1/4
Explanation: Step 1: n(S) = 2 × 6 = 12 equally likely outcomes. Step 2: X ≥ 6 only for (H, 5), (H, 6) and (T, 6). ∴ P(X ≥ 6) = 3/12 = 1/4 (2/7 is a trap: it treats the values 1, …, 7 of the range as equally likely).
If X is a discrete random variable whose probability distribution is xᵣ: 1, 2, 3, 4 and f(xᵣ): 0.2, 0.3, k, m, and the two events “X ≤ 2” and “X is an even number” are independent, then k = ……
Correct answer: (A) 0.2
Explanation: Step 1: k + m = 1 − 0.5 = 0.5. P(X ≤ 2) = 0.5, P(X is even) = 0.3 + m, and their intersection is “X = 2” with probability 0.3. Step 2: For independent events the probability of the intersection = the product of the probabilities: 0.3 = 0.5 × (0.3 + m), which gives m = 0.3 ∴ k = 0.5 − 0.3 = 0.2 (0.3 is the value of m, not k).
In a game, a regular die is rolled: the player wins 3 pounds if 6 appears, wins 1 pound if 4 or 5 appears, and loses 2 pounds otherwise. The player plays twice, and X expresses his net gain from the two games. Then P(X > 0) = ……
Correct answer: (D) 5/12
Explanation: Step 1: In each game: P(+3) = 1/6, P(+1) = 1/3, P(−2) = 1/2. Step 2: X > 0 if he does not lose in either game: (1/2)² = 1/4, or he wins 3 and loses 2 (X = 1) in either order: 2 × (1/6) × (1/2) = 1/6. ∴ P(X > 0) = 1/4 + 1/6 = 5/12 (1/4 is a trap: it forgets the second case).
X expresses the number of Heads when a fair coin is tossed 3 times, and Y expresses the number of girls in a family with two children (the probability of a boy = the probability of a girl). Which of the following statements is true?
Correct answer: (B) P(X ≥ 1) > P(Y ≥ 1)
Explanation: Step 1: The distribution of X: xᵣ: 0, 1, 2, 3 and f(xᵣ): 1/8, 3/8, 3/8, 1/8. The distribution of Y: 0, 1, 2 with probabilities 1/4, 1/2, 1/4. Step 2: P(X ≥ 1) = 7/8, which is greater than P(Y ≥ 1) = 3/4. ∴ P(X ≥ 1) > P(Y ≥ 1) (but P(X = 1) = 3/8 ≠ 1/2, P(X = 0) = 1/8 ≠ 1/4, and P(X ≤ 1) = 1/2 is less than P(Y ≤ 1) = 3/4).
The first 10 questions in every subject are free, each with an explanation.