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.
-
Q1 Medium
P(A and B) = 0.12, P(B) = 0.4. What is P(A|B)?
ExplanationP(A|B) = 0.12/0.4 = 0.3. -
Q2 Easy
Fair coin tossed twice. Given first is head, what is P(second is head)?
ExplanationIndependent: P = 1/2. -
Q3 Medium
Bag: 4 red, 1 blue. One ball drawn and it is red. Two balls total drawn without replacement — given first red, P(second red)?
ExplanationAfter one red removed: 3 red, 1 blue left. P = 3/4. -
Q4 Medium
Bag: 3 red, 2 blue. One draw without replacement. Given first is blue, P(second is red)?
ExplanationAfter blue removed: 3 red, 1 blue. P = 3/4. -
Q5 Medium
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Are A and B independent?
Explanation0.5 × 0.4 = 0.2, so independent. -
Q6 Hard
Standard deck. One card drawn. Given it is red, what is P(king)?
Explanation2 red kings out of 26 red cards: P = 2/26 = 1/13. -
Q7 Hard
Two fair dice rolled. Given sum is 7, what is P(first die shows 2)?
ExplanationGiven sum 7, six equally likely pairs; one has first die 2: P = 1/6.
-
Q8 Medium
P(A|B) = 0.6 and P(B) = 0.5. What is P(A and B)?
ExplanationP(A and B) = P(A|B) × P(B) = 0.6 × 0.5 = 0.3. -
Q9 Easy
Bag: 2 red, 3 blue. One ball drawn. Given it is not blue, what is P(red)?
ExplanationNot blue means red in this bag: P = 1. -
Q10 Medium
P(A and B) = 0.15, P(A) = 0.3. What is P(B|A)?
ExplanationP(B|A) = 0.15/0.3 = 0.5. -
Q11 Hard
Two cards drawn without replacement. Given first is ace, P(second is ace)?
Explanation3 aces remain out of 51 cards: P = 3/51. -
Q12 Medium
Fair die rolled. Given number is even, what is P(greater than 4)?
ExplanationGiven even {2,4,6}, only 6 > 4: P = 1/3. -
Q13 Medium
P(A|B) = P(A). What does this suggest?
ExplanationP(A|B) = P(A) is the independence condition. -
Q14 Hard
Bag: 5 white, 5 black. Two draws without replacement. Given first is white, P(second black)?
ExplanationAfter one white removed: 5 white, 5 black. P = 5/9.
-
DS1 Medium
What is P(A|B)?
Statement IP(A and B) = 0.18.
Statement IIP(B) = 0.6.
ExplanationNeed both: P(A|B) = 0.18/0.6 = 0.3. -
DS2 Easy
What is P(second head given first head on two fair coins)?
Statement ITwo fair coins are tossed.
Statement IIThe tosses are independent.
ExplanationStatement II alone: P = 1/2.
-
DS3 Hard
What is P(second red given first red without replacement)?
Statement IA bag has 3 red and 2 blue balls.
Statement IITwo balls are drawn without replacement and the first is red.
ExplanationNeed both: after one red, 2 red of 4 remain: P = 2/4 = 1/2. -
DS4 Hard
What is P(king given red card)?
Statement IThere are 2 red kings among 26 red cards.
Statement IIThe drawn card is known to be red.
ExplanationNeed both: P = 2/26 = 1/13.