Practice Questions
Concept Explanation
Mixed problems combine basic probability, complements, independence, mutually exclusive events, and conditional reasoning in one scenario.
Strategy
- Identify the sample space and whether outcomes are equally likely
- Choose direct counting, complement, multiplication, or addition as needed
- Check with replacement vs without replacement
Example
Two dice: P(sum at least 10) = P(10)+P(11)+P(12) = 3/36 + 2/36 + 1/36 = 1/6.
Common Trap
Mixing OR and AND rules incorrectly is the main error in combined problems.
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Q1 Medium
Two dice rolled. What is P(sum at least 10)?
ExplanationSums 10,11,12: 3+2+1 = 6 outcomes. P = 6/36 = 1/6. -
Q2 Medium
Three coins tossed. What is P(at least two heads)?
ExplanationHHH, HHT, HTH, THH: P = 4/8 = 1/2. -
Q3 Hard
One card drawn. What is P(king or heart)?
Explanation13 hearts + 3 non-heart kings = 16 cards. P = 16/52 = 4/13. -
Q4 Hard
Bag: 4 red, 6 blue. Two draws with replacement. What is P(one red and one blue)?
ExplanationRB or BR: 2 × (4/10)(6/10) = 48/100 = 12/25. -
Q5 Medium
Two fair dice. What is P(sum is 7 or 11)?
Explanation6 + 2 = 8 outcomes. P = 8/36 = 2/9. -
Q6 Hard
P(A) = 0.5, P(B) = 0.4, independent. P(A or B)?
ExplanationP = 0.5 + 0.4 − 0.2 = 0.7. -
Q7 Medium
Three fair coins. What is P(exactly two tails)?
ExplanationTTH, THT, HTT: P = 3/8.
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Q8 Hard
Bag: 2 red, 3 blue, 5 white. Two draws without replacement. What is P(both same colour)?
ExplanationRR: (2/10)(1/9); BB: (3/10)(2/9); WW: (5/10)(4/9). Sum = 28/90 = 14/45. -
Q9 Medium
Two dice rolled. What is P(sum less than 6)?
ExplanationSums 2,3,4,5: 1+2+3+4 = 10 outcomes. P = 10/36 = 5/18. -
Q10 Hard
One card drawn. What is P(black or king)?
Explanation26 black + 2 red kings = 28 cards. P = 28/52 = 7/13. -
Q11 Medium
Four fair coins. What is P(exactly three heads)?
Explanation4 ways: P = 4/16 = 1/4. -
Q12 Medium
Bag: 3 red, 2 blue. Two draws without replacement. What is P(different colours)?
ExplanationRB or BR: 2 × (3/5)(2/4) = 6/10 = 3/5. -
Q13 Hard
Two dice rolled. What is P(product is 12)?
ExplanationPairs (2,6),(3,4),(4,3),(6,2): P = 4/36 = 1/9. -
Q14 Medium
P(E) = 0.6. Two independent trials same experiment. P(at least one E)?
Explanation1 − 0.4^2 = 1 − 0.16 = 0.84.
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DS1 Medium
What is P(sum at least 10 when two dice are rolled)?
Statement ITwo fair dice are rolled.
Statement IIThere are 6 outcomes with sum at least 10.
ExplanationNeed both: P = 6/36 = 1/6. -
DS2 Medium
What is P(at least two heads when three coins are tossed)?
Statement IThree fair coins are tossed.
Statement IIExactly two or three heads are counted as success.
ExplanationNeed both: 4 favourable of 8.
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DS3 Hard
What is P(king or heart on one card draw)?
Statement IOne card is drawn from a standard deck.
Statement IIHearts and kings overlap by one card.
ExplanationNeed both: P = 16/52 = 4/13. -
DS4 Hard
What is P(A or B) for independent events?
Statement IP(A) = 0.5 and P(B) = 0.4.
Statement IIA and B are independent.
ExplanationNeed both: P = 0.5 + 0.4 − 0.2 = 0.7.