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.
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Q1 Easy
Two fair coins tossed. Are the two tosses independent?
ExplanationEach toss does not affect the other. -
Q2 Medium
Fair coin and fair die used. P(head and 4)?
ExplanationP = (1/2) × (1/6) = 1/12. -
Q3 Easy
Two fair dice rolled. P(both show 3)?
ExplanationP = (1/6) × (1/6) = 1/36. -
Q4 Easy
P(A) = 0.4, P(B) = 0.5, A and B independent. P(A and B)?
ExplanationP = 0.4 × 0.5 = 0.2. -
Q5 Easy
Two fair coins. P(first head and second tail)?
ExplanationP = (1/2) × (1/2) = 1/4. -
Q6 Easy
Bag: draw with replacement each time. Are two draws independent?
ExplanationReplacement restores the bag each time. -
Q7 Medium
P(A) = 1/3, P(B) = 3/4, independent. P(A and B)?
ExplanationP = (1/3) × (3/4) = 3/12 = 1/4.
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Q8 Medium
Three fair coins. P(all three heads)?
ExplanationP = (1/2)^3 = 1/8. -
Q9 Medium
Two fair dice. P(first even and second odd)?
ExplanationP = (1/2) × (1/2) = 1/4. -
Q10 Medium
Machine A works with P = 0.9, B with P = 0.8, independent. P(both work)?
ExplanationP = 0.9 × 0.8 = 0.72. -
Q11 Hard
Two cards drawn with replacement from deck. P(both aces)?
ExplanationP = (4/52) × (4/52) = (1/13)^2 = 1/169. -
Q12 Hard
P(A) = 0.6, P(B) = 0.5, independent. P(neither A nor B)?
ExplanationP = (1−0.6) × (1−0.5) = 0.4 × 0.5 = 0.2. -
Q13 Hard
Fair die rolled twice. P(both rolls greater than 4)?
ExplanationP(each > 4) = 2/6 = 1/3. P = (1/3)^2 = 1/9. -
Q14 Easy
Spinner P(red) = 0.3, coin P(H) = 0.5, independent. P(red and head)?
ExplanationP = 0.3 × 0.5 = 0.15.
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DS1 Easy
What is P(A and B) if A and B are independent?
Statement IP(A) = 0.3.
Statement IIP(B) = 0.4 and A, B are independent.
ExplanationNeed both: P = 0.3 × 0.4 = 0.12. -
DS2 Medium
What is P(both heads when two fair coins are tossed)?
Statement IEach fair coin has P(head) = 1/2.
Statement IIThe two tosses are independent.
ExplanationNeed both: P = (1/2) × (1/2) = 1/4.
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DS3 Medium
What is P(head on coin and 4 on die)?
Statement IP(head) = 1/2 on a fair coin.
Statement IIP(getting 4) = 1/6 on a fair die and the events are independent.
ExplanationNeed both: P = (1/2) × (1/6) = 1/12. -
DS4 Easy
What is P(both dice show 6)?
Statement ITwo fair dice are rolled.
Statement IIThe two rolls are independent.
ExplanationNeed both: P = (1/6) × (1/6) = 1/36.