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 101 questions on this lesson, each with an explanation.
If A is a certain event in the sample space S of a random experiment, then P(A) = ……
Correct answer: (A) 1
Explanation: Step 1: The certain event is S itself (its elements are the elements of S). Step 2: By the axioms of probability, P(S) = 1 ∴ P(A) = 1 (0 is the probability of the impossible event φ.)
When a coin is tossed three consecutive times and the sequence of heads and tails is observed, n(S) = ……
Correct answer: (B) 8
Explanation: Step 1: Tossing a coin m times gives n(S) = 2ᵐ. Step 2: n(S) = 2³ = 2 × 2 × 2 ∴ n(S) = 8 (6 = 2 × 3 and 9 = 3² mix up the base and the power.)
In a school of 150 students, each student practises exactly one activity. Boys: 40 sports, 25 art, 15 music. Girls: 20 sports, 35 art, 15 music. A student is chosen at random. The probability that the student is a girl who practises art = ……
Correct answer: (A) 7/30
Explanation: Step 1: Number of girls who practise art = 35, out of n(S) = 150 Step 2: P = 35/150 ∴ P = 7/30 (1/2 = 35/70 divides by the number of girls only; 2/5 = 60/150 is the probability of practising art; 7/15 = 70/150 is the probability of being a girl.)
A restaurant offers 3 starters (soup, salad, pickles), 2 main dishes (fish, meat) and 2 desserts (cake, fruit). A person chooses a meal with one item of each kind at random. The probability that the meal contains soup or cake = ……
Correct answer: (D) 2/3
Explanation: Step 1: n(S) = 3 × 2 × 2 = 12 equally likely meals Step 2: Meals with neither soup nor cake = 2 × 2 × 1 = 4, so by De Morgan P(soup or cake) = 1 − 4/12 ∴ P = 2/3 (or 4/12 + 6/12 − 2/12) (5/6 = 1/3 + 1/2 forgets to subtract the probability of the intersection, 1/6.)
A coin is tossed three consecutive times. The probability of getting at least one head and at least one tail = ……
Correct answer: (C) 3/4
Explanation: Step 1: The complementary event is getting the same face in the three tosses: (H, H, H) or (T, T, T), with probability 2/8 Step 2: P(A) = 1 − P(A') ∴ P = 1 − 2/8 = 3/4 (7/8 is the probability of at least one head only; 1/4 is the probability of the complement.)
A regular die is rolled two consecutive times. The probability that the absolute difference between the two numbers equals 1 = ……
Correct answer: (A) 5/18
Explanation: Step 1: n(S) = 6 × 6 = 36 Step 2: The event: (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), (5, 6), (6, 5), 10 outcomes ∴ P = 10/36 = 5/18 (5/36 is a trap: it counts the pairs in one direction only, but (1, 2) and (2, 1) are different outcomes.)
From the digits {0, 1, 2, 3}, a two-digit number with two different digits is formed (the tens digit is not zero), and one of these numbers is chosen at random. The probability that the number is even = ……
Correct answer: (C) 5/9
Explanation: Step 1: The tens digit is 1, 2 or 3 (3 ways) and the unit digit is any different digit (3 ways), so n(S) = 9: 10, 12, 13, 20, 21, 23, 30, 31, 32 Step 2: Even numbers: 10, 12, 20, 30, 32, 5 numbers ∴ P = 5/9 (1/2 = 6/12 is a trap: it allows 0 as the tens digit (01, 02, …); 4/9 is the probability of an odd number.)
If A and B are two events in the sample space of a random experiment, P(A') = (2/3)P(A), P(B) = 0.5 and P(A' ∪ B') = 0.7, then the probability of occurrence of only one of the two events = ……
Correct answer: (D) 0.5
Explanation: Step 1: 1 − P(A) = (2/3)P(A), so (5/3)P(A) = 1 and P(A) = 0.6 Step 2: P(A' ∪ B') = P((A ∩ B)') = 0.7, so P(A ∩ B) = 0.3; P(A ∪ B) = 0.6 + 0.5 − 0.3 = 0.8 Step 3: Only one of the two events = P(A ∪ B) − P(A ∩ B) = 0.8 − 0.3 ∴ = 0.5 (0.8 is the probability of one of the two events at least.)
The probability that a student succeeds in the physics exam is 0.7, the probability that he succeeds in the mathematics exam is 0.8, and the probability that he fails both exams is 0.1. The probability that he succeeds in both exams = ……
Correct answer: (C) 0.6
Explanation: Step 1: Let A = success in physics and B = success in mathematics. Failing both = A' ∩ B' = (A ∪ B)', so P(A ∪ B) = 1 − 0.1 = 0.9 Step 2: P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.7 + 0.8 − 0.9 ∴ P(A ∩ B) = 0.6 (0.56 = 0.7 × 0.8 is a trap: multiplying is not justified here; 0.9 is the probability of success in at least one exam.)
A box contains identical white and red balls. If a ball is drawn at random, the probability that it is red is 1/4. If 4 red balls are added to the box, the probability of drawing a red ball becomes 1/3. The number of white balls in the box = ……
Correct answer: (A) 24
Explanation: Step 1: Let the number of red balls be r and the total be n: r/n = 1/4, so n = 4r Step 2: (r + 4)/(n + 4) = 1/3, so 3r + 12 = 4r + 4 and r = 8, n = 32 ∴ white balls = 32 − 8 = 24 (adding red balls does not change it) (32 is the total number of balls.)
The first 10 questions in every subject are free, each with an explanation.