How to Solve Basic Conditional Probability Problems
Learn conditional probability with the shrink-the-sample-space method, the two-child family trap, and worked examples with answers.
Expected Interview Answer
Conditional probability P(A|B) = P(A and B) / P(B) measures the chance of event A given that event B has already happened, which shrinks the sample space from everything down to just the outcomes where B is true.
The key mental shift is that B becomes the new, smaller universe — instead of dividing by all outcomes, you divide by only the outcomes consistent with B. In a two-child-family style problem, 'given that at least one child is a boy' eliminates the all-girls case, leaving a reduced sample space over which you recount favorable outcomes. Conditional probability is distinct from independence: A and B are independent exactly when P(A|B) = P(A), meaning knowing B happened gives no information about A. Most interview-level conditional problems are solved by explicitly listing the reduced sample space rather than blindly plugging into the formula, since listing prevents subtle miscounting.
- Reframing B as the new sample space avoids formula misapplication
- Explicit listing catches classic miscounting traps (like the two-child problem)
- Distinguishing independence from conditioning clarifies when P(A|B) simplifies to P(A)
AI Mentor Explanation
Asking 'what is the chance a randomly chosen wicket is a bowled dismissal' is unconditional, using all wickets in the sample space. Asking 'what is the chance it was bowled, given it happened in the last 5 overs’ is conditional — you first shrink the sample space to only last-5-over wickets, then find how many of those were bowled, dividing by that smaller total rather than all wickets in the match. That shrink-then-recount is exactly P(A|B) = P(A and B)/P(B).
Worked example (two-child family)
Full sample space
- {BB, BG, GB, GG}
Given: at least one boy
- Shrinks to {BB, BG, GB}
P(both boys | at least one boy)
- 1/3
Step-by-Step Explanation
Step 1
Identify A and B
A is the event asked about; B is the given condition.
Step 2
Shrink the sample space to B
List or count only outcomes consistent with B having occurred.
Step 3
Count favorable outcomes within B
Find how many of B's outcomes also satisfy A.
Step 4
Divide within the shrunk space
P(A|B) = (outcomes satisfying A and B) / (outcomes satisfying B).
What Interviewer Expects
- Correct formula P(A|B) = P(A and B)/P(B)
- Explicit listing of the reduced sample space to avoid miscounting
- Clear distinction between conditional probability and independence
- Recognition of classic traps like the two-child family problem
Common Mistakes
- Dividing by the full sample space instead of the reduced one given B
- Assuming P(A|B) = P(A) without checking independence
- Miscounting the two-child problem as 1/2 instead of 1/3
- Confusing P(A|B) with P(B|A) — these are generally different values
Best Answer (HR Friendly)
“Conditional probability means the given information shrinks my sample space — I stop dividing by everything and start dividing only by the outcomes consistent with what I am told already happened. I find that explicitly listing the reduced sample space, rather than jumping straight to the formula, avoids classic traps like the two-child family problem.”
Follow-up Questions
- How does P(A|B) differ from P(B|A)?
- When does P(A|B) equal P(A), and what does that tell you about A and B?
- How would you extend conditional probability to three events?
- How does Bayes' theorem build on conditional probability?
MCQ Practice
1. A family has two children. Given at least one is a girl, what is P(both are girls)?
Sample space {BB,BG,GB,GG} shrinks to {BG,GB,GG} given at least one girl; favorable GG is 1 of 3.
2. If P(A and B) = 0.2 and P(B) = 0.5, what is P(A|B)?
P(A|B) = P(A and B)/P(B) = 0.2/0.5 = 0.4.
3. If P(A|B) = P(A), what can be concluded about A and B?
When knowing B occurred does not change the probability of A, A and B are independent by definition.
Flash Cards
Conditional probability formula? — P(A|B) = P(A and B) / P(B).
What does “given B” do to the sample space? — Shrinks it to only outcomes where B is true.
Independence test using conditional probability? — A and B are independent iff P(A|B) = P(A).
Classic conditional probability trap? — Two-child family: P(both boys | at least one boy) = 1/3, not 1/2.