Practice Questions
Concept Explanation
Two events are independent if one does not affect the other. The probability of both occurring is the product of their individual probabilities.
Key Formula
- P(A and B) = P(A) × P(B) when A and B are independent
- Fair coin tosses and die rolls with replacement are independent
- Drawing without replacement creates dependence
Example
Two fair coins: P(both heads) = 1/2 × 1/2 = 1/4.
Common Trap
Do not multiply unless events are truly independent.
-
Q1 EasyTwo fair coins are tossed independently. What is P(first head and second tail)?ExplanationP = (1/2) × (1/2) = 1/4. Hence the answer is 1/4.
-
Q2 EasyA and B are independent events with P(A) = 0.4 and P(B) = 0.5. What is P(A and B)?ExplanationFor independent events, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.2. Hence the answer is 0.2.
-
Q3 EasyTwo fair dice are rolled. What is P(both show 3)?ExplanationIndependent rolls: P = (1/6) × (1/6) = 1/36. Hence the answer is 1/36.
-
Q4 EasyA and B are two independent events. If P(A) = 1/3 and P(B) = 1/4, then P(A and B) is equal to:ExplanationIndependence ⇒ P(A ∩ B) = P(A) × P(B) = (1/3) × (1/4) = 1/12. Hence the answer is 1/12.
-
Q5 MediumIn a campus quiz, a fair coin and a fair die are used independently. What is P(head and 4)?ExplanationIndependence ⇒ P(H and 4) = P(H) × P(4) = (1/2) × (1/6) = 1/12. Hence the answer is 1/12.
-
Q6 MediumTwo fair dice are thrown. What is the probability that the first die shows an even number and the second die shows an odd number?ExplanationP(even on first) = 3/6 = 1/2 and P(odd on second) = 3/6 = 1/2. The rolls are independent, so P = (1/2) × (1/2) = 1/4. Hence the answer is 1/4.
-
Q7 MediumTwo bags each contain 3 white and 2 black balls. One ball is drawn from each bag. What is the probability that both balls drawn are black?ExplanationDraws from separate bags are independent. P(black from one bag) = 2/5. P(both black) = (2/5) × (2/5) = 4/25. Hence the answer is 4/25.
-
Q8 MediumA fresher paper states: P(A) = 1/3, P(B) = 3/4, and A, B are independent. What is P(A and B)?ExplanationP(A ∩ B) = (1/3) × (3/4) = 3/12 = 1/4. Hence the answer is 1/4.
-
Q9 MediumThree fair coins are tossed independently. What is P(all three heads)?ExplanationP = (1/2)^3 = 1/8. Hence the answer is 1/8.
-
Q10 MediumMachine A works with probability 0.9 and machine B with 0.8; the machines work independently. What is P(both work)?ExplanationP(both) = 0.9 × 0.8 = 0.72. Hence the answer is 0.72.
-
Q11 MediumA bag contains 4 white and 6 black balls. Two balls are drawn one after another with replacement. What is P(both are black)?ExplanationWith replacement, draws are independent: P(black) = 6/10 = 3/5 each time. P(both black) = (3/5) × (3/5) = 9/25. Hence the answer is 9/25.
-
Q12 HardTwo cards are drawn with replacement from a pack of 52. What is P(both aces)?ExplanationWith replacement the draws are independent: P = (4/52) × (4/52) = (1/13)^2 = 1/169. Hence the answer is 1/169.
-
Q13 HardIf A and B are independent events with P(A) = 0.6 and P(B) = 0.5, what is P(neither A nor B)?ExplanationP(neither) = P(A′)P(B′) = (1 − 0.6)(1 − 0.5) = 0.4 × 0.5 = 0.2. Hence the answer is 0.2.
-
Q14 HardA fair die is rolled twice independently. What is P(both rolls are greater than 4)?ExplanationFaces greater than 4 are 5 and 6, so P = 2/6 = 1/3 each roll. P(both) = (1/3) × (1/3) = 1/9. Hence the answer is 1/9.
-
DS1 Easy
In a campus quiz, what is P(A and B) if A and B are independent?
Statement IP(A) = 0.3.
Statement IIP(B) = 0.4 and it is given that A and B are independent.
ExplanationI alone lacks P(B). II alone lacks P(A). Together: P(A ∩ B) = 0.3 × 0.4 = 0.12. Answer: C. -
DS2 Medium
What is P(first die even and second die odd)?
Statement ITwo fair dice are rolled independently.
Statement IIP(even on a fair die) = 1/2 and P(odd) = 1/2.
ExplanationI alone needs the even/odd probabilities. II alone needs that both dice are rolled independently. Together: P = (1/2)×(1/2) = 1/4. Answer: C.
-
DS3 Medium
What is P(both balls drawn are black from two identical bags)?
Statement IEach bag has 3 white and 2 black balls, and one ball is drawn from each bag.
Statement IIThe two draws are independent.
ExplanationStatement I alone: drawing one ball from each separate bag gives independent draws, so P = (2/5)×(2/5) = 4/25. Statement II alone lacks the bag composition. Answer: A. -
DS4 Hard
What is P(neither A nor B) for independent events A and B?
Statement IP(A) = 0.6 and P(B) = 0.5.
Statement IIA and B are independent.
ExplanationI alone lacks independence (needed to multiply complements). II alone lacks the probabilities. Together: P = 0.4 × 0.5 = 0.2. Answer: C.