Unit 3 — Probability · 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 85 questions on this lesson, each with an explanation.
In an experiment of tossing a regular coin once and then rolling a regular die, the probability of getting a head and a number greater than 4 equals ……
Correct answer: (A) 1/6
Explanation: Step 1: The two events are independent: P(head) = 1/2 and P(number greater than 4) = 2/6 = 1/3 Step 2: P = 1/2 × 1/3 = 1/6 ∴ The probability = 1/6 (1/12 is a trap: it takes the number 5 only)
If A and B are two independent events and P(B) ≠ 0, then P(A | B) = ……
Correct answer: (A) P(A)
Explanation: Step 1: P(A | B) = P(A ∩ B) ÷ P(B) = P(A) × P(B) ÷ P(B) Step 2: = P(A) ∴ P(A | B) = P(A): the occurrence of B does not affect the probability of A
A bag contains 3 white balls and 7 black balls. A ball is drawn at random, put back in the bag, and then a second ball is drawn. The probability that the two balls are white equals ……
Correct answer: (A) 9/100
Explanation: Step 1: The drawing is with replacement, so the two events are independent Step 2: P = 3/10 × 3/10 = 9/100 ∴ The probability = 9/100 (1/15 = 3/10 × 2/9 is for drawing without replacement)
Two soldiers A and B each fire a missile at a target, and each soldier's hit is independent of the other's. If the probability that soldier A hits the target is 0.6 and the probability that soldier B hits it is 0.5, then the probability that the target is not hit equals ……
Correct answer: (C) 0.2
Explanation: Step 1: Let A and B be the events that soldiers A and B hit the target. The target is not hit if both miss: P(A' ∩ B') = P(A') × P(B') Step 2: P(A' ∩ B') = 0.4 × 0.5 = 0.2 ∴ The probability = 0.2 (0.3 is the probability that both hit; 0.8 is the probability that the target is hit; 0.7 = 1 − 0.3)
If A and B are two independent events, each with probability between 0 and 1, which of the following pairs of events is also independent?
Correct answer: (C) A' and B'
Explanation: Step 1: P(A' ∩ B') = 1 − P(A ∪ B) = 1 − P(A) − P(B) + P(A) × P(B) Step 2: = (1 − P(A))(1 − P(B)) = P(A') × P(B'), so A' and B' are independent ∴ The pair is A' and B' (A and A' are mutually exclusive with positive probabilities, so they are dependent)
The probability that the first machine in a factory works a full day without a breakdown is 0.8, and for the second machine it is 0.9. The two machines work independently. The probability that at most one machine works a full day without a breakdown equals ……
Correct answer: (B) 0.28
Explanation: Step 1: At most one machine is the complement of both machines working: P(both) = 0.8 × 0.9 = 0.72 Step 2: P(at most one) = 1 − 0.72 = 0.28 ∴ The probability = 0.28 (0.26 is the probability of exactly one machine; 0.02 is the probability of neither)
If A and B are two mutually exclusive events of a sample space S where P(A) > 0 and P(B) > 0, then the two events A and B are ……
Correct answer: (B) dependent
Explanation: Step 1: A and B are mutually exclusive, so P(A ∩ B) = 0, while P(A) × P(B) > 0 Step 2: So P(A ∩ B) ≠ P(A) × P(B) ∴ A and B are dependent (like getting a head and a tail in one toss of a coin)
A device has three components that work independently, with probabilities of working 0.9, 0.8 and 0.5 respectively. The probability that exactly two of the three components work equals ……
Correct answer: (D) 0.49
Explanation: Step 1: Exactly two = (first and second, not third) + (first and third, not second) + (second and third, not first) Step 2: = 0.9 × 0.8 × 0.5 + 0.9 × 0.2 × 0.5 + 0.1 × 0.8 × 0.5 = 0.36 + 0.09 + 0.04 = 0.49 ∴ The probability = 0.49 (0.36 is the probability that all three work; 0.85 is at least two; 0.14 is exactly one)
If A and B are two independent events where P(A) = 0.5 and P(B) = 0.4, then the probability that A occurs given that at least one of the two events has occurred, that is P(A | A ∪ B), equals ……
Correct answer: (A) 5/7
Explanation: Step 1: P(A ∩ B) = 0.5 × 0.4 = 0.2 (independent), so P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7 Step 2: A ⊂ (A ∪ B), so A ∩ (A ∪ B) = A and P(A | A ∪ B) = 0.5 ÷ 0.7 = 5/7 ∴ P(A | A ∪ B) = 5/7 (1/2 ignores the condition; 3/7 is the probability of A only under the same condition)
Two students, Ali and Omar, try to solve a problem, each independently of the other. The probability that both solve it is 0.3 and the probability that the problem is solved is 0.8. If Ali is more likely than Omar to solve it, then the probability that Ali solves it equals ……
Correct answer: (B) 0.6
Explanation: Step 1: Let the two probabilities be x and y: x × y = 0.3 (independent) and x + y − 0.3 = 0.8, so x + y = 1.1 Step 2: The two numbers with sum 1.1 and product 0.3 are 0.6 and 0.5; the larger is Ali's ∴ P(Ali solves it) = 0.6 (0.5 is Omar's probability; 0.55 is half the sum)
The first 10 questions in every subject are free, each with an explanation.