Practice Questions
Concept Explanation
Probability DS items test whether favorable and total outcomes can be counted uniquely.
Standard Answer Choices
- A — Statement I alone is sufficient; Statement II alone is not.
- B — Statement II alone is sufficient; Statement I alone is not.
- C — Both together are sufficient, neither alone.
- D — Either statement alone is sufficient.
- E — Neither alone nor together is sufficient.
Probability DS Focus
P(E) = Favorable ÷ Total. Independent events multiply; mutually exclusive events add.
DS Decision Flow
- State exactly what must be determined (number, ratio, yes/no, or comparison).
- Test Statement I alone—sufficient or insufficient?
- Test Statement II alone—sufficient or insufficient?
- If neither alone works, combine both statements.
- Choose A–E based on sufficiency; never pick a value when data do not fix it uniquely.
Common Traps in Probability Problems
- Assuming events are independent without statement support.
- Double-counting overlapping outcomes.
- Confusing “at least one” with simple addition.
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Q1 EasyFor “P(red)?” — I: 3 red balls; II: 8 total balls. DS answer?ExplanationP = 3/8. Need both counts. Hence the answer is C.
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Q2 EasyFair die rolled once. P(getting 4) =ExplanationOne favorable out of six outcomes. Hence the answer is 1/6.
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Q3 EasyBag: 3 red, 5 blue balls. P(red) =Explanation3/(3+5) = 3/8. Hence the answer is 3/8.
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Q4 EasyTwo fair coins tossed. P(both heads) =Explanation1 of 4 outcomes: HH. Hence the answer is 1/4.
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Q5 EasyCard drawn from 52-card deck. P(king) =Explanation4 kings in 52 cards = 1/13. Hence the answer is 1/13.
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Q6 EasyFor “Number of blue balls?” — I: total 10; II: red = 3. DS answer?ExplanationBlue = 10 − 3 = 7 only with both (assuming only red and blue). Hence the answer is C.
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Q7 EasyFor “Is P(A) > 1/2?” — I: P(A) = 0.6; II: favorable = 6, total = 10. DS answer?ExplanationEach gives P = 0.6 > 0.5. Hence the answer is D.
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Q8 EasyP(impossible event) =ExplanationAn impossible event has probability 0. Hence the answer is 0.
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Q9 MediumFor “P(even on standard die)?” — I: die has faces 1–6; II: exactly three outcomes are even. DS answer?ExplanationStatement I alone: 3 even out of 6 ⇒ P = 1/2. Statement II alone lacks total outcomes. Hence the answer is A.
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Q10 MediumP(at least one head in two coin tosses) =Explanation1 − P(no heads) = 1 − 1/4 = 3/4. Hence the answer is 3/4.
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Q11 MediumFor “P(both independent events occur)?” — I: P(A)=0.4; II: P(B)=0.5. DS answer?ExplanationP(A and B) = 0.4 × 0.5 = 0.2 if independent. Need both probabilities. Hence the answer is C.
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Q12 MediumTwo dice rolled. P(sum = 7) =Explanation6 favorable pairs out of 36. Hence the answer is 1/6.
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Q13 HardFor “P(drawing red)?” — I: 4 red, 6 blue; II: 2 green also present. DS answer?ExplanationI alone: 4/10. II alone lacks red count. Together: 4/12 = 1/3. Hence the answer is C.
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Q14 HardBox has 5 white and 3 black balls. Two drawn with replacement. P(both white) =Explanation(5/8)² = 25/64. Hence the answer is 25/64.
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DS1 Easy
What is the probability of drawing a red ball?
Statement IThere are 3 red balls in the bag.
Statement IIThere are 8 balls in total.
ExplanationP = 3/8. Need both red count and total. Answer: C. -
DS2 Easy
What is the probability of getting an even number on a fair die?
Statement IThe die has faces 1 through 6.
Statement IIExactly three outcomes are even.
ExplanationStatement I alone: 3 even out of 6 ⇒ P = 1/2. Statement II alone does not fix total outcomes. Answer: A.
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DS3 Easy
How many blue balls are in the bag (only red and blue)?
Statement ITotal balls = 10.
Statement IIRed balls = 3.
ExplanationBlue = 10 − 3 = 7. Need both counts. Answer: C. -
DS4 Medium
What is P(A and B) if A and B are independent?
Statement IP(A) = 0.4.
Statement IIP(B) = 0.5.
ExplanationP(A and B) = 0.4 × 0.5 = 0.2. Need both probabilities. Answer: C.