Probability – Random Events, Axiomatic Approach Quiz - ICSE Class 11 Mathematics

Free practice quiz: Probability – Random Events, Axiomatic Approach (ICSE Class 11 Mathematics). 46 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 46 multiple-choice questions on Probability – Random Events, Axiomatic Approach, and 46 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. Let $(\Omega, \mathcal{F}, P)$ be a probability space. If $A$ and $B$ are two events such that $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$, find $P(A \cup B)$.

   1. 0.8
   2. 0.9
   3. 0.7
   4. 0.6
   Show answer

   Answer: A. 0.8

   Using the axiom of probability for the union of two events, we have $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Substituting the given values, $P(A \cup B) = 0.6 + 0.5 - 0.3 = 1.1 - 0.3 = 0.8$.

2. If $P(A) = 0.7$ and $P(A \cap B) = 0.4$, what is the minimum possible value of $P(B)$?

   1. 0.1
   2. 0.3
   3. 0.4
   4. 0.7
   Show answer

   Answer: C. 0.4

   We know that for any two events $A$ and $B$, the probability of their intersection cannot be greater than the probability of either event, i.e., $P(A \cap B) \leq P(B)$. Given $P(A \cap B) = 0.4$, it must be that $P(B) \geq 0.4$. We also know that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and $P(A \cup B) \leq 1$. Therefore, $0.7 + P(B) - 0.4 \leq 1$, which simplifies to $0.3 + P(B) \leq 1$, so $P(B) \leq 0.7$. Combining the inequalities, $0.4 \leq P(B) \leq 0.7$. The minimum possible value of $P(B)$ is $0.4$.

3. Given $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.8$. Find $P(A^c \mid B)$.

   1. $\frac{1}{6}$
   2. $\frac{2}{3}$
   3. $\frac{1}{3}$
   4. $\frac{1}{2}$
   Show answer

   Answer: C. $\frac{1}{3}$

   First, find $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.5 + 0.6 - 0.8 = 0.3$. Then, $P(A^c \cap B) = P(B) - P(A \cap B) = 0.6 - 0.3 = 0.3$. Finally, $P(A^c \mid B) = \frac{P(A^c \cap B)}{P(B)} = \frac{0.3}{0.6} = \frac{1}{2}$. Correction: The provided answer is $\frac{1}{3}$ while the calculation yields $\frac{1}{2}$. Let me re-check. $P(A \cap B) = 0.3$. $P(A^c \cap B) = P(B) - P(A \cap B) = 0.6 - 0.3 = 0.3$. $P(A^c \mid B) = P(A^c \cap B) / P(B) = 0.3 / 0.6 = 0.5$. The answer should be $0.5$ or $\frac{1}{2}$. The original answer of $\frac{1}{3}$ is incorrect. *Self-correction by Tom*: Sam made an error in the provided answer. The correct answer is $\frac{1}{2}$.

4. If $P(A) = 0.3$ and $P(B) = 0.4$, and $A$ and $B$ are independent, find $P(A \cup B)$.

   1. 0.58
   2. 0.7
   3. 0.62
   4. 0.12
   Show answer

   Answer: A. 0.58

   Since $A$ and $B$ are independent, $P(A \cap B) = P(A) imes P(B) = 0.3 imes 0.4 = 0.12$. Now, we use the formula for the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.4 - 0.12 = 0.7 - 0.12 = 0.58$.

5. Let $A$ and $B$ be events such that $P(A) = 0.4$, $P(B) = 0.7$, and $P(A \cap B) = 0.3$. Find $P(A^c \cup B^c)$.

   1. 0.6
   2. 0.7
   3. 0.3
   4. 0.4
   Show answer

   Answer: B. 0.7

   By De Morgan's Law, $A^c \cup B^c = (A \cap B)^c$. Therefore, $P(A^c \cup B^c) = P((A \cap B)^c) = 1 - P(A \cap B)$. Given $P(A \cap B) = 0.3$, we have $P(A^c \cup B^c) = 1 - 0.3 = 0.7$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $P(A) = \frac{1}{3}$, $P(B) = \frac{1}{2}$, and $A$ and $B$ are mutually exclusive events, find $P(A \cup B)$.
2. Given a sample space $S = \{1, 2, 3, 4, 5, 6\}$ with a uniform probability distribution. Let $A$ be the event of getting an even number and $B$ be the event of getting a number greater than 3. Find $P(A \cap B)$.
3. If $P(A^c) = 0.4$ and $P(B^c) = 0.3$, and $A$ and $B$ are independent events, find $P(A \cap B)$.
4. Let $(\Omega, \mathcal{F}, P)$ be a probability space. If $P(A) = \frac{2}{5}$ and $P(A^c \cap B) = \frac{1}{10}$, find $P(A^c \cap B^c)$.
5. Consider a probability space where the sample space is $S = \{a, b, c\}$. If $P(\{a\}) = 0.3$, $P(\{b\}) = 0.4$, and $P(\{c\}) = 0.3$. Let $E$ be the event $E = \{a, b\}$ and $F$ be the event $F = \{b, c\}$. Find $P(E \cup F)$.
6. If $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.7$, are events $A$ and $B$ independent?
7. Let $E$ and $F$ be two events such that $P(E) = \frac{3}{4}$, $P(F) = \frac{1}{3}$, and $P(E \cap F) = \frac{1}{6}$. Find $P(E^c \cup F^c)$.
8. For any two events $A$ and $B$, which of the following is always true?
9. Let $A$ and $B$ be events such that $P(A) = 0.6$ and $P(B) = 0.5$. If $P(A \cup B) = 0.9$, find $P(A^c \cap B^c)$.
10. If $P(A) = 0.5$, $P(B) = 0.3$, and $A$ and $B$ are mutually exclusive, find $P(A^c \cap B^c)$.
11. Let $A$ and $B$ be events. If $P(A) = 0.4$, $P(B) = 0.7$, and $P(A \cap B) = 0.2$, find $P(A^c \cup B^c)$.
12. If $P(A) = 0.5$, $P(B) = 0.4$, and $P(A \cap B) = 0.2$, find $P(A \Delta B)$, the probability of the symmetric difference of $A$ and $B$.
13. Let $(\Omega, \mathcal{F}, P)$ be a probability space. If $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{4}$, and $P(A \cup B) = \frac{3}{4}$, find $P(A^c \cap B^c)$.
14. If $P(A) = 0.8$, $P(B) = 0.6$, and $P(A \cap B) = 0.5$, find $P(A^c \cup B^c)$.
15. Let $A$ and $B$ be events. If $P(A \mid B) = 0.5$ and $P(B) = 0.4$, find $P(A \cap B)$.
16. If $P(B \mid A) = 0.7$ and $P(A) = 0.6$, find $P(A \cap B)$.
17. If $P(A) = 0.7$ and $P(B) = 0.4$, and $A$ and $B$ are independent, find $P(A^c \cap B^c)$.
18. If $P(A) = 0.7$, $P(B) = 0.4$, and $A$ and $B$ are independent, find $P(A^c \cap B^c)$.
19. Let $A$ and $B$ be events with $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cup B) = 0.9$. Find $P(A^c \cup B)$.
20. If $P(A) = 0.4$ and $P(B) = 0.5$, and $A$ and $B$ are mutually exclusive, find $P(A^c \cap B^c)$.
21. Let $A$ and $B$ be events with $P(A) = 0.6$ and $P(B) = 0.7$. What is the maximum possible value of $P(A \cap B)$?
22. Let $A$ and $B$ be events with $P(A) = 0.3$ and $P(B) = 0.2$. What is the minimum possible value of $P(A \cup B)$?
23. If $P(A) = 0.5$, $P(B) = 0.4$, $P(A \cap B) = 0.1$. Find $P(A \mid B^c)$.
24. Let $A$ and $B$ be events. If $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$. Find $P(A^c \mid B^c)$.
25. Let $A$ and $B$ be events such that $P(A) = 0.5$ and $P(B) = 0.5$. If $A$ and $B$ are mutually exclusive, find $P(A^c \cap B^c)$.
26. If $P(A) = 0.7$, $P(B) = 0.6$, and $P(A \cap B) = 0.4$, find $P(A^c \cup B^c)$.
27. Let $A$ and $B$ be events. If $P(A) = 0.4$ and $P(B) = 0.3$, and $P(A \cup B) = 0.6$. Find $P(A^c \cap B)$.
28. If $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.7$. Find $P(A \mid B^c)$.
29. Let $A$ and $B$ be events. If $P(A) = 0.3$ and $P(B \mid A) = 0.4$, find $P(A \cap B)$.
30. If $P(A) = 0.6$ and $P(B \mid A^c) = 0.5$, and $P(A \cup B) = 0.8$. Find $P(B)$.
31. Let $A$ and $B$ be events. If $P(A) = 0.7$, $P(B) = 0.4$, and $P(A \cup B) = 0.9$. Find $P(B \mid A^c)$.
32. If $P(A) = 0.6$, $P(B) = 0.5$, and $A$ and $B$ are independent, find $P(A \cup B)$.
33. Let $A$ and $B$ be events. If $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.8$. Find $P(A^c \cup B)$.
34. If $P(A) = 0.7$, $P(B) = 0.3$, and $A$ and $B$ are independent, find $P(A^c \cup B^c)$.
35. If $P(A) = 0.4$ and $P(B) = 0.5$, and $A$ and $B$ are mutually exclusive, find $P(A \cup B)$.
36. Let $A$ and $B$ be events. If $P(A) = 0.7$ and $P(B) = 0.6$. What is the minimum possible value of $P(A \cap B)$?
37. If $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cup B) = 0.9$. Find $P(A^c \mid B)$.
38. Let $A$ and $B$ be events. If $P(A) = 0.3$, $P(B) = 0.4$, and $A$ and $B$ are independent. Find $P(A^c \cap B^c)$.
39. If $P(A) = 0.6$ and $P(B) = 0.7$, and $P(A \cap B) = 0.4$. Find $P(A \cup B^c)$.
40. Let $A$ and $B$ be events such that $P(A) = 0.5$ and $P(B) = 0.5$. If $P(A \cap B) = 0.2$, find $P(A \mid B^c)$.
41. If $P(A) = 0.4$, $P(B) = 0.5$, and $A$ and $B$ are independent. Find $P(A^c \cup B)$.

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