How to Solve Probability Problems
Solve probability aptitude problems — favorable/total, AND/OR rules, complement and conditional probability — with a worked example and practice questions.
Expected Interview Answer
Probability of an event is favorable outcomes divided by total possible outcomes, P(E) = n(E)/n(S), and most problems are solved by carefully counting both using permutation/combination techniques, then applying independence, mutual exclusivity, or conditional rules as needed.
For independent events, probabilities multiply: P(A and B) = P(A) × P(B). For mutually exclusive events, probabilities add: P(A or B) = P(A) + P(B). When events are not mutually exclusive, use P(A or B) = P(A) + P(B) − P(A and B) to avoid double-counting the overlap. Conditional probability P(A|B) = P(A and B)/P(B) handles cases where one event’s outcome affects another’s, common in "without replacement" draws. The complement rule, P(not A) = 1 − P(A), is often the fastest route for "at least one" problems.
- One core formula (favorable/total) anchors every variation
- The complement rule shortcuts "at least one" problems dramatically
- AND/OR rules cover independence and overlap correctly
AI Mentor Explanation
A bowler with a 70% chance of taking a wicket each over — asking the chance of taking at least one wicket in 3 overs is much easier via the complement: 1 minus the chance of zero wickets in all three overs (0.3×0.3×0.3), rather than adding up every possible combination of successes directly. Probability problems favor exactly this complement trick, plus multiplying independent-event probabilities (each over’s outcome is independent of the last) rather than adding them.
Worked example (without replacement)
First draw
- P(red) = 4/10
Second draw
- P(red | first red) = 3/9
Combined
- 4/10 × 3/9 = 2/15
Step-by-Step Explanation
Step 1
Define the sample space
Count total possible outcomes n(S), often via permutation/combination.
Step 2
Count favorable outcomes
Count n(E) matching the event, using the same counting method.
Step 3
Apply AND/OR rules
Independent AND multiplies; mutually exclusive OR adds; overlapping OR subtracts the intersection.
Step 4
Use the complement when useful
For "at least one," compute 1 − P(none) instead of enumerating cases.
What Interviewer Expects
- Correct favorable/total setup using counting principles
- Correct application of AND (independent) vs OR (mutually exclusive/overlapping) rules
- Recognizing when the complement rule simplifies "at least one" problems
- Correct conditional probability for draws without replacement
Common Mistakes
- Adding probabilities of non-mutually-exclusive events without subtracting the overlap
- Multiplying probabilities of dependent events as if independent
- Forgetting to update the sample space after a draw without replacement
- Enumerating "at least one" cases instead of using the complement rule
Best Answer (HR Friendly)
“Probability is just favorable outcomes over total outcomes, and the hard part is counting both correctly. Multiply probabilities for independent events, add them for mutually exclusive ones, and subtract the overlap when they can happen together. For "at least one" questions, it is almost always faster to compute one minus the probability of none happening.”
Follow-up Questions
- How do you compute probability for events that are not independent?
- When would you use Bayes’ theorem in an interview-style problem?
- How does "with replacement" change a draw problem versus "without replacement"?
- How do you compute the probability of exactly k successes in n trials?
MCQ Practice
1. A fair coin is tossed 3 times. The probability of getting at least one head is?
P(no heads) = (1/2)^3 = 1/8; P(at least one head) = 1 − 1/8 = 7/8.
2. A bag has 3 red and 7 blue balls. One ball is drawn at random. P(red) is?
Favorable outcomes (red) = 3, total = 10, so P(red) = 3/10.
3. Two dice are rolled. The probability that the sum is 7 is?
6 favorable pairs out of 36 total outcomes: 6/36 = 1/6.
Flash Cards
Core probability formula? — P(E) = favorable outcomes ÷ total outcomes.
AND rule (independent events)? — P(A and B) = P(A) × P(B).
OR rule (overlapping events)? — P(A or B) = P(A) + P(B) − P(A and B).
Complement rule use case? — "At least one" problems: P(at least one) = 1 − P(none).