Practice Questions
Concept Explanation
Conditional probability P(A|B) is the probability of A given that B has already occurred.
Key Formula
- P(A|B) = P(A and B) / P(B), when P(B) > 0
- For independent events: P(A|B) = P(A)
- Without replacement, later draws depend on earlier results
Example
Bag: 3 red, 2 blue. Draw red then (without replacement) red: P = (3/5) × (2/4) = 3/10.
Common Trap
Update the sample space after the given condition.
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Q1 EasyP(A ∩ B) = 0.12 and P(B) = 0.4. What is P(A|B)?ExplanationBy definition, P(A|B) = P(A ∩ B) / P(B) = 0.12 / 0.4 = 0.3. Hence the answer is 0.3.
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Q2 EasyTwo fair coins are tossed. Given that the first coin shows head, what is P(second coin shows head)?ExplanationThe tosses are independent, so P(second head | first head) = P(second head) = 1/2. Hence the answer is 1/2.
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Q3 MediumA card is drawn from a pack of 52 cards and is found to be red. What is the probability that it is a heart?ExplanationGiven the card is red, the reduced sample space has 26 red cards. Hearts among red = 13. P(heart | red) = 13/26 = 1/2. Using the formula: P(heart ∩ red)/P(red) = (13/52)/(26/52) = 1/2. Hence the answer is 1/2.
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Q4 MediumA bag contains 4 red and 5 blue balls. Two balls are drawn one by one without replacement. Given that the first ball drawn is red, what is the probability that the second ball drawn is blue?ExplanationAfter drawing one red, 3 red and 5 blue remain (8 balls). P(second blue | first red) = 5/8. Hence the answer is 5/8.
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Q5 MediumA bag has 3 red and 2 blue balls. Two are drawn without replacement. Given the first is blue, what is P(second is red)?ExplanationAfter one blue is removed, 3 red and 1 blue remain. P = 3/4. Hence the answer is 3/4.
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Q6 MediumAn HR quiz states P(A|B) = 0.6 and P(B) = 0.5. What is P(A ∩ B)?ExplanationRearrange: P(A ∩ B) = P(A|B) × P(B) = 0.6 × 0.5 = 0.3. Hence the answer is 0.3.
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Q7 MediumA die is thrown. Given that the number is even, what is the probability that it is 6?ExplanationGiven even, the sample space is {2,4,6}. Only one favourable face is 6. P(6 | even) = 1/3. Hence the answer is 1/3.
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Q8 MediumP(A ∩ B) = 0.15 and P(A) = 0.3. What is P(B|A)?ExplanationP(B|A) = P(A ∩ B) / P(A) = 0.15 / 0.3 = 0.5. Hence the answer is 0.5.
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Q9 MediumA die is thrown. Given that the outcome is even, what is the probability that it is a multiple of 3?ExplanationEven faces: {2,4,6}. Multiples of 3 among them: only 6. P = 1/3. Hence the answer is 1/3.
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Q10 HardA family has two children. It is known that at least one of them is a boy. What is the probability that both children are boys? (Assume boy/girl equally likely and independent.)ExplanationEqually likely outcomes: BB, BG, GB, GG. Given at least one boy, the sample space is {BB, BG, GB}. Exactly one of these three is BB. P(both boys | at least one boy) = 1/3. Hence the answer is 1/3.
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Q11 HardTwo fair dice are rolled. Given that the sum is 7, what is P(the first die shows 2)?ExplanationOutcomes with sum 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — six equally likely pairs. Exactly one has first die 2. P = 1/6. Hence the answer is 1/6.
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Q12 HardTwo cards are drawn without replacement. Given the first is an ace, what is P(second is an ace)?ExplanationAfter one ace is removed, 3 aces remain among 51 cards. P = 3/51. Hence the answer is 3/51.
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Q13 HardA card is drawn from a pack of 52 cards and is known to be red. What is the probability that it is a king?ExplanationGiven red, there are 26 red cards including 2 red kings. P(king | red) = 2/26 = 1/13. Hence the answer is 1/13.
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Q14 HardA bag has 5 white and 5 black balls. Two are drawn without replacement. Given the first is white, what is P(second is black)?ExplanationAfter one white is removed, 4 white and 5 black remain (9 balls). P = 5/9. Hence the answer is 5/9.
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DS1 Medium
In a campus conditional-probability drill, what is P(A|B)?
Statement IP(A ∩ B) = 0.18.
Statement IIP(B) = 0.6.
ExplanationNeed both: P(A|B) = 0.18 / 0.6 = 0.3. Answer: C. -
DS2 Medium
A placement worksheet asks: a die shows an even number. What is P(it is 6)?
Statement IA fair six-sided die is used.
Statement IIThe outcome is known to be even.
ExplanationTogether: given {2,4,6}, P(6) = 1/3. Answer: C.
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DS3 Hard
What is P(second red | first red) without replacement?
Statement IA bag has 3 red and 2 blue balls.
Statement IITwo balls are drawn without replacement and the first drawn is red.
ExplanationTogether: after one red removed, 2 red of 4 remain, so P = 2/4 = 1/2. Answer: C. -
DS4 Hard
A drawn card is known to be red. What is P(it is a king)?
Statement IThere are 2 red kings among 26 red cards.
Statement IIThe drawn card is known to be red.
ExplanationNeed both: P(king | red) = 2/26 = 1/13. Answer: C.