Probability – Conditional, Bayes, Random Variables & Distributions Quiz - ICSE Class 12 Mathematics

Free practice quiz: Probability – Conditional, Bayes, Random Variables & Distributions (ICSE Class 12 Mathematics). 25 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 12 Mathematics practice quiz has 25 multiple-choice questions on Probability – Conditional, Bayes, Random Variables & Distributions, and 25 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Two events A and B are such that P(A) = 1/2, P(B) = 1/3, and P(A ∩ B) = 1/4. What is P(A' ∪ B')?

   1. 1/4
   2. 3/4
   3. 1/2
   4. 5/6
   Show answer

   Answer: B. 3/4

   We know that P(A' ∪ B') = P((A ∩ B)'). Using the complement rule, P((A ∩ B)') = 1 - P(A ∩ B). Given P(A ∩ B) = 1/4, so P(A' ∪ B') = 1 - 1/4 = 3/4.

2. Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what is P(A ∪ B)?

   1. 0.7
   2. 0.9
   3. 0.2
   4. 1.0
   Show answer

   Answer: A. 0.7

   For independent events, P(A ∩ B) = P(A) * P(B). So, P(A ∩ B) = 0.4 * 0.5 = 0.2. The formula for P(A ∪ B) is P(A) + P(B) - P(A ∩ B). Therefore, P(A ∪ B) = 0.4 + 0.5 - 0.2 = 0.7.

3. Suppose P(E) = 0.4 and P(F) = 0.3. If E and F are mutually exclusive events, what is P(E ∪ F)?

   1. 0.1
   2. 0.7
   3. 0.12
   4. 0.3
   Show answer

   Answer: B. 0.7

   For mutually exclusive events, P(E ∩ F) = 0. The formula for P(E ∪ F) is P(E) + P(F) - P(E ∩ F). Therefore, P(E ∪ F) = 0.4 + 0.3 - 0 = 0.7.

4. Two events A and B are such that P(A|B) = 0.7 and P(B|A) = 0.4. If P(A) = 0.5, find P(B).

   1. 0.286
   2. 0.35
   3. 0.571
   4. 0.2
   Show answer

   Answer: A. 0.286

   We know P(A|B) = P(A ∩ B) / P(B) and P(B|A) = P(A ∩ B) / P(A). From the second equation, P(A ∩ B) = P(B|A) * P(A) = 0.4 * 0.5 = 0.2. Now substitute this into the first equation: 0.7 = 0.2 / P(B). Therefore, P(B) = 0.2 / 0.7 = 2/7. Approximately, 2/7 ≈ 0.2857, which rounds to 0.286.

5. Suppose P(A) = 0.3 and P(B) = 0.4. If events A and B are mutually exclusive, what is P(A ∩ B)?

   1. 0
   2. 0.1
   3. 0.7
   4. 0.12
   Show answer

   Answer: A. 0

   For mutually exclusive events, the occurrence of one event excludes the occurrence of the other. Therefore, the probability of both events occurring simultaneously, P(A ∩ B), is 0.

More questions in this quiz

Answers and explanations are in the full quiz.

1. A fair die is tossed twice. Let X be the number of sixes obtained. Find the probability distribution of X.
2. Given P(A) = 0.6, P(B) = 0.7, and P(A ∩ B) = 0.4. What is P(A|B)?
3. A bag contains 3 red balls and 2 blue balls. Two balls are drawn without replacement. What is the probability that the first ball is red and the second ball is blue?
4. Let X be a discrete random variable with the following probability distribution: P(X=1) = 0.2, P(X=2) = 0.3, P(X=3) = 0.5. Calculate the expected value of X.
5. Bayes' Theorem states that P(A|B) = [P(B|A) * P(A)] / P(B). If P(A) = 0.3, P(B) = 0.6, and P(B|A) = 0.8, what is P(A|B)?
6. A random variable X follows a Poisson distribution with mean λ = 3. What is the probability that X = 4?
7. If P(A) = 0.5, P(B) = 0.4, and P(A ∪ B) = 0.7, are events A and B independent?
8. A bag contains 5 red and 5 blue balls. Three balls are drawn at random without replacement. What is the probability that all three balls are red?
9. A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?
10. Suppose X ~ B(n, p). If E(X) = 6 and Var(X) = 3, find n and p.
11. In a medical test, the probability of a false positive is 0.05 and the probability of a false negative is 0.10. If the prevalence of the disease is 0.02, what is the probability that a person tests positive given they have the disease?
12. Let X be a random variable representing the number of heads in 3 coin flips. What is P(X=2)?
13. A student takes a multiple-choice test with 10 questions, each with 4 options. If the student guesses randomly, what is the probability that they answer exactly 3 questions correctly?
14. A fair coin is tossed 10 times. What is the expected number of heads?
15. A random variable X has a probability density function f(x) = 2x for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. What is P(0.2 ≤ X ≤ 0.5)?
16. Let X be a random variable with a uniform distribution on the interval [0, 10]. What is the probability that X > 7?
17. Two events A and B are such that P(A) = 0.6, P(B) = 0.5, and P(A ∪ B) = 0.7. What is P(A ∩ B)?
18. Let A and B be two events. If P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, what is P(A' ∪ B')?
19. A random variable X follows a Poisson distribution with mean λ. If P(X=0) = 1/e, what is the value of λ?
20. Suppose P(A) = 0.4 and P(B) = 0.3. If events A and B are independent, what is P(A' ∩ B')?

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