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 EasyA card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is:ExplanationThese are two specific distinct cards. Favourable = 2. P = 2/52 = 1/26. Hence the answer is 1/26.
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Q2 EasyIn a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?ExplanationNeither red nor green means blue. Favourable = 7, total = 8 + 7 + 6 = 21. P = 7/21 = 1/3. Hence the answer is 1/3.
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Q3 MediumTwo dice are thrown. What is the probability that the sum is at least 10?ExplanationFavourable outcomes for sum ≥ 10: (4,6),(5,5),(6,4),(5,6),(6,5),(6,6) — 6 outcomes. P = 6/36 = 1/6. Hence the answer is 1/6.
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Q4 MediumThree fair coins are tossed. What is P(exactly two tails)?ExplanationFavourable: TTH, THT, HTT — 3 of 8. P = 3/8. Hence the answer is 3/8.
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Q5 MediumTwo fair dice are rolled. What is P(sum is 7 or 11)?ExplanationSum 7 has 6 outcomes; sum 11 has 2. Total favourable = 8. P = 8/36 = 2/9. Hence the answer is 2/9.
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Q6 MediumTwo dice are tossed. The probability that the product of the numbers on the two dice is even is:ExplanationProduct is odd only if both faces are odd: 3 × 3 = 9 outcomes. P(odd product) = 9/36 = 1/4. P(even product) = 1 − 1/4 = 3/4. Hence the answer is 3/4.
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Q7 MediumBag has 3 red and 2 blue. Two draws without replacement. What is P(different colours)?ExplanationRB or BR: (3/5)(2/4) + (2/5)(3/4) = 6/20 + 6/20 = 12/20 = 3/5. Hence the answer is 3/5.
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Q8 MediumFour fair coins are tossed. What is P(exactly three heads)?ExplanationNumber of ways = C(4,3) = 4. Total = 16. P = 4/16 = 1/4. Hence the answer is 1/4.
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Q9 MediumA fair coin is tossed and a fair die is thrown. What is the probability of getting a head and an even number?ExplanationThe coin and die are independent. P(head) = 1/2 and P(even) = 1/2. P(both) = (1/2) × (1/2) = 1/4. Hence the answer is 1/4.
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Q10 HardP(A) = 0.5, P(B) = 0.4 and A, B are independent. What is P(A or B)?ExplanationP(A ∩ B) = 0.5 × 0.4 = 0.2. P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7. Hence the answer is 0.7.
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Q11 HardA bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?ExplanationTotal ways = C(7,2) = 21. Non-blue balls = 2 + 3 = 5. Ways = C(5,2) = 10. P = 10/21. Hence the answer is 10/21.
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Q12 HardAn HR quiz states: one card is drawn. What is P(black or king)?ExplanationBlack cards = 26; red kings = 2. Favourable = 26 + 2 = 28. P = 28/52 = 7/13. Hence the answer is 7/13.
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Q13 HardA bag has 4 red and 6 blue balls. Two draws are made with replacement. What is P(one red and one blue in some order)?ExplanationRB or BR: 2 × (4/10) × (6/10) = 48/100 = 12/25. Hence the answer is 12/25.
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Q14 HardFrom a pack of 52 cards two cards are drawn together at random. What is the probability of both the cards being kings?ExplanationTotal ways = C(52,2) = 1326. Favourable = C(4,2) = 6. P = 6/1326 = 1/221. Hence the answer is 1/221.
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DS1 Easy
What is P(queen of clubs or king of hearts on one draw)?
Statement IOne card is drawn from a standard 52-card deck.
Statement IIThese are two distinct specific cards.
ExplanationTogether: P = 2/52 = 1/26. Answer: C. -
DS2 Medium
In a campus mixed-probability drill, what is P(sum at least 10 when two dice are rolled)?
Statement ITwo fair dice are rolled.
Statement IIThere are exactly 6 outcomes with sum at least 10.
ExplanationNeed both: P = 6/36 = 1/6. Answer: C.
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DS3 Medium
A placement worksheet asks: a coin is tossed and a die is thrown. What is P(head and even)?
Statement IThe coin is fair (P(head) = 1/2).
Statement IIThe die is fair and the two experiments are independent.
ExplanationTogether: P = (1/2) × (1/2) = 1/4. Answer: C. -
DS4 Hard
For P(A ∪ B) when A and B are independent. Can it be found?
Statement IP(A) = 0.5 and P(B) = 0.4.
Statement IIA and B are independent.
ExplanationTogether: P(A ∩ B) = 0.2, so P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7. Answer: C.