Practice Questions
Concept Explanation
An event is a set of outcomes from a sample space. The probability of an event sums favourable equally likely outcomes and divides by the total.
Key Rules
- Sample space S = all possible outcomes
- P(E) = n(E) / n(S) when outcomes are equally likely
- P(E) + P(not E) = 1
Example
Two dice rolled: 36 outcomes. Event sum = 7 has 6 outcomes. P(sum 7) = 6/36 = 1/6.
Common Trap
Define the sample space first. Rolling two dice has 36 ordered outcomes, not 11 sums.
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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:ExplanationP(E) = favourable / total. Favourable cards: queen of clubs and king of hearts → 2 distinct cards. Total = 52. P = 2/52 = 1/26. Hence the answer is 1/26.
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Q2 EasyTwo fair dice are rolled. How many outcomes are in the sample space?ExplanationEach die has 6 faces, so ordered outcomes n(S) = 6 × 6 = 36. Hence the answer is 36.
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Q3 EasyA card is drawn from a standard deck. What is the probability that it is a king?ExplanationP(E) = favourable / total. Favourable kings = 4, total = 52. P = 4/52 = 1/13. Hence the answer is 1/13.
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Q4 EasyFrom digits 1, 2, 3, 4, 5 one digit is chosen at random. What is the probability of an even digit?ExplanationP(E) = favourable / total. Even digits: 2, 4 → 2 favourable out of 5. P = 2/5. Hence the answer is 2/5.
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Q5 EasyOne card is drawn from a deck. What is the probability that it is a red card?ExplanationP(E) = favourable / total. Red cards = 26, total = 52. P = 26/52 = 1/2. Hence the answer is 1/2.
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Q6 EasyA die is thrown. What is the probability of getting an odd number?ExplanationP(E) = favourable / total. Odd faces: 1, 3, 5 → 3 out of 6. P = 3/6 = 1/2. Hence the answer is 1/2.
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Q7 EasyThree fair coins are tossed. How many outcomes are in the sample space?ExplanationEach coin has 2 outcomes, so n(S) = 2^3 = 8. Hence the answer is 8.
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Q8 MediumTwo dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?ExplanationP(E) = favourable / total. Total outcomes = 36. Product is odd only if both faces are odd: odd faces {1,3,5} → 3 × 3 = 9. Favourable (even product) = 36 − 9 = 27. P = 27/36 = 3/4. Hence the answer is 3/4.
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Q9 MediumA letter is chosen at random from the word PROBABILITY. What is the probability that the letter is a vowel?ExplanationP(E) = favourable / total. Letters in PROBABILITY = 11. Vowels O, A, I, I → 4. P = 4/11. Hence the answer is 4/11.
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Q10 MediumThree unbiased coins are tossed. What is the probability of getting at least two heads?ExplanationP(E) = favourable / total. Sample space = 2^3 = 8. At least two heads: HHH, HHT, HTH, THH → 4. P = 4/8 = 1/2. Hence the answer is 1/2.
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Q11 MediumTwo dice are rolled. What is the probability that the sum equals 8?ExplanationP(E) = favourable / total. Sum 8: (2,6),(3,5),(4,4),(5,3),(6,2) → 5 outcomes. P = 5/36. Hence the answer is 5/36.
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Q12 MediumTwo coins are tossed. The probability of getting at least one head is:ExplanationP(E) = favourable / total. Outcomes: HH, HT, TH, TT → 4. At least one head: HH, HT, TH → 3. P = 3/4. Equivalently 1 − P(TT) = 1 − 1/4 = 3/4. Hence the answer is 3/4.
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Q13 HardTwo dice are tossed. The probability that the total score is a prime number is:ExplanationP(E) = favourable / total. Total = 36. Prime totals: 2, 3, 5, 7, 11. Counts: sum 2 → 1; sum 3 → 2; sum 5 → 4; sum 7 → 6; sum 11 → 2. Favourable = 1 + 2 + 4 + 6 + 2 = 15. P = 15/36 = 5/12. Hence the answer is 5/12.
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Q14 HardTwo dice are rolled. What is the probability that the sum is at least 11?ExplanationP(E) = favourable / total. Sum ≥ 11 means sum 11 or 12: (5,6),(6,5),(6,6) → 3 outcomes. P = 3/36 = 1/12. Hence the answer is 1/12.
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DS1 Easy
What is the probability of drawing a king from a deck?
Statement IOne card is drawn at random from a deck of 52 cards.
Statement IIThere are exactly 4 kings in the deck.
ExplanationStatement I alone lacks the number of kings. Statement II alone lacks the deck size. Together: P = 4/52 = 1/13. Both needed. Answer: C. -
DS2 Easy
How many outcomes are in the sample space when two dice are rolled. What is that number?
Statement IEach die has 6 faces.
Statement IIOutcomes are counted as ordered pairs (first die, second die).
ExplanationStatement I alone does not confirm ordered vs unordered counting. Statement II alone lacks the face count. Together: 6 × 6 = 36. Both needed. Answer: C.
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DS3 Medium
What is the probability that the sum equals 7 when two dice are rolled?
Statement ITwo fair dice are rolled, giving 36 equally likely outcomes.
Statement IIExactly 6 of the outcomes have sum equal to 7.
ExplanationStatement I alone lacks the favourable count. Statement II alone lacks the total sample-space size. Together: P = 6/36 = 1/6. Both needed. Answer: C. -
DS4 Medium
For the probability that the sum is even when two dice are rolled. What is that probability?
Statement IThere are 36 equally likely ordered outcomes and exactly 18 have an even sum.
Statement IIExactly half of the 36 equally likely outcomes have an even sum.
ExplanationStatement I alone: P = 18/36 = 1/2. Statement II alone: P = 1/2. Each alone is sufficient. Answer: D.