Practice Questions
Concept Explanation
Drawing balls from a bag uses P(E) = favourable / total when each ball is equally likely. With replacement, totals stay fixed; without replacement, totals shrink.
Key Rules
- With replacement: probabilities stay the same each draw
- Without replacement: adjust counts after each draw
- Two draws without replacement: multiply sequential probabilities
Example
Bag: 3 red, 2 blue. P(red) = 3/5. Without replacement, P(second red | first red) = 2/4.
Common Trap
Check whether drawing is with or without replacement before computing.
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Q1 EasyIn a campus drill, a bag has 5 white and 3 black balls. One ball is drawn at random. What is P(black)?ExplanationFavourable = 3 black, total = 8. P(black) = 3/8. Hence the answer is 3/8.
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Q2 EasyA bag has 4 red and 6 blue balls. One ball is drawn. What is P(red)?ExplanationP(red) = 4/10 = 2/5. Hence the answer is 2/5.
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Q3 EasyA bag has 3 red, 3 blue and 3 green balls. One ball is drawn. What is P(not green)?ExplanationNon-green = 6, total = 9. P = 6/9 = 2/3. Hence the answer is 2/3.
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Q4 MediumA 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 balls = 7; non-blue = 2 + 3 = 5. Without replacement, P = C(5,2)/C(7,2) = 10/21. Sequentially: (5/7)×(4/6) = 20/42 = 10/21. Hence the answer is 10/21.
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Q5 MediumA bag contains 4 white and 6 black balls. Two balls are drawn at random. What is the probability that both balls are white?ExplanationWithout replacement: P = C(4,2)/C(10,2) = 6/45 = 2/15. Or (4/10)×(3/9) = 12/90 = 2/15. Hence the answer is 2/15.
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Q6 MediumA bag has 3 red and 2 blue balls. Two balls are drawn without replacement. What is P(both red)?ExplanationP = (3/5)×(2/4) = 6/20 = 3/10. Equivalently C(3,2)/C(5,2) = 3/10. Hence the answer is 3/10.
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Q7 MediumA bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that both are white is:ExplanationWithout replacement: P = C(6,2)/C(10,2) = 15/45 = 1/3. Or (6/10)×(5/9) = 30/90 = 1/3. Hence the answer is 1/3.
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Q8 MediumA bag has 7 red and 3 blue balls. Two balls are drawn with replacement. What is P(both red)?ExplanationWith replacement the bag is restored: P = (7/10)×(7/10) = 49/100. Hence the answer is 49/100.
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Q9 MediumIn an online test, a bag has 5 red and 5 blue balls. Two draws are made with replacement. What is P(first red, second blue)?ExplanationWith replacement: P = (5/10)×(5/10) = (1/2)×(1/2) = 1/4. Hence the answer is 1/4.
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Q10 MediumA bag contains 4 white and 6 black balls. Two balls are drawn one after another with replacement. What is the probability that both are white?ExplanationWith replacement: P(first white) = 4/10 and P(second white) = 4/10. P(both white) = (4/10)×(4/10) = 16/100 = 4/25. Hence the answer is 4/25.
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Q11 MediumA bag contains 4 white and 6 black balls. Two balls are drawn without replacement. What is the probability that both are black?ExplanationWithout replacement: P = C(6,2)/C(10,2) = 15/45 = 1/3. Or (6/10)×(5/9) = 30/90 = 1/3. Hence the answer is 1/3.
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Q12 HardA bag has 2 red, 3 blue and 5 white balls. Two balls are drawn without replacement. What is P(first red, second blue)?ExplanationTotal = 10. P = (2/10)×(3/9) = 6/90 = 1/15. Hence the answer is 1/15.
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Q13 HardBag has 6 red and 2 blue. Two draws with replacement. What is P(at least one blue)?ExplanationUse complement: P(no blue) = (6/8)×(6/8) = 36/64. P(at least one blue) = 1 − 36/64 = 28/64 = 7/16. Hence the answer is 7/16.
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Q14 HardA bag contains 6 white and 4 black balls. Two balls are drawn with replacement. What is the probability that both balls drawn are white?ExplanationWith replacement: P = (6/10)×(6/10) = 36/100 = 9/25. (Contrast without replacement, which would be 1/3.) Hence the answer is 9/25.
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DS1 Easy
In a campus bag problem, what is P(drawing a red ball)?
Statement IThe bag has 5 red and 5 blue balls.
Statement IIThere are 10 balls in total and one ball is drawn at random.
ExplanationStatement I alone: 5 red out of 10 ⇒ P(red) = 1/2. Statement II alone gives the total but not how many are red. Answer: A. -
DS2 Medium
What is P(both draws are white with replacement)?
Statement IThe bag has 4 white and 6 black balls.
Statement IITwo balls are drawn with replacement.
ExplanationI alone does not say with/without replacement. II alone lacks the colour counts. Together: P = (4/10)^2 = 4/25. Answer: C.
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DS3 Medium
What is P(both white without replacement from a bag of white and black balls)?
Statement IThe bag has 6 white and 4 black balls.
Statement IITwo balls are drawn without replacement.
ExplanationTogether: P = (6/10)×(5/9) = 1/3. Neither alone is enough. Answer: C. -
DS4 Hard
A bag has 2 red, 3 green and 2 blue balls. What is P(none of two drawn balls is blue)?
Statement ITwo balls are drawn at random without replacement.
Statement IINon-blue balls number 5 out of 7.
ExplanationI alone needs non-blue count; II alone needs that two are drawn without replacement. Together: C(5,2)/C(7,2) = 10/21. Answer: C.