Explaining Probability in Interviews

Explaining to a Non-Technical Interviewer

When asked to explain probability concepts, lead with the intuition, then the formula:

"What is conditional probability?"

"Conditional probability answers the question: given that something has already happened, how likely is something else? For example: the probability that a patient has AF is 5% in the general population. But the probability that a patient has AF given that they're already on warfarin is much higher — maybe 60% — because warfarin is often prescribed for AF. We write this as P(AF | warfarin), and it equals the fraction of warfarin patients who have AF."

"What is the difference between joint and marginal probability?"

"Joint probability is the probability that two things are both true — like the probability that someone both has diabetes AND has high blood pressure. It accounts for how these conditions relate. Marginal probability is the probability of just one thing being true, without caring about the other — the probability that someone has diabetes, regardless of whether they have high blood pressure. You get the marginal by 'summing over' all the joint possibilities — adding up the probability of 'diabetes + high BP' and 'diabetes + no high BP'."

Common Interview Questions

Q1: "What is P(A or B)?"

"P(A ∪ B) = P(A) + P(B) - P(A ∩ B). We subtract P(A ∩ B) because if we just add P(A) and P(B), we double-count the overlap — cases where both A and B occur. For mutually exclusive events where A and B can't both happen, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B) simply."

# Example: probability of rolling a 1 OR an even number
Omega = {1, 2, 3, 4, 5, 6}
A = {1}          # P(A) = 1/6
B = {2, 4, 6}    # P(B) = 3/6
# A ∩ B = {} (mutually exclusive — 1 is not even)
# P(A ∪ B) = 1/6 + 3/6 - 0 = 4/6 = 0.667

Q2: "If I flip two fair coins, what's P(at least one head)?"

"Use the complement. P(at least one head) = 1 - P(no heads). P(no heads) = P(both tails) = 0.5 × 0.5 = 0.25. So P(at least one head) = 1 - 0.25 = 0.75. The complement trick is often faster than summing up P(exactly one head) + P(exactly two heads)."

# Analytical
p_no_heads = 0.5 * 0.5  # both tails
p_at_least_one = 1 - p_no_heads  # 0.75

# Simulation
import numpy as np
n_trials = 100_000
flips = np.random.choice([0, 1], size=(n_trials, 2))  # 0=tail, 1=head
p_estimated = np.mean(flips.sum(axis=1) >= 1)
print(f"Analytical: 0.75, Simulated: {p_estimated:.4f}")

Q3: "What does it mean for two events to be independent?"

"A and B are independent if knowing that A occurred tells you nothing about whether B occurred. Formally: P(A | B) = P(A), or equivalently P(A ∩ B) = P(A) × P(B). In practice: the result of the first coin flip doesn't change the probability of the second — they're independent. Whether someone has AF and whether they have a broken arm are probably independent. But having AF and being on warfarin are NOT independent — AF causes warfarin use, so they're correlated."

Q4: "What's the probability of drawing two aces from a deck without replacement?"

"Without replacement means the second draw depends on the first — this is conditional probability. P(first ace) = 4/52. Given the first card was an ace, there are now 3 aces left in 51 remaining cards: P(second ace | first ace) = 3/51. Using the multiplication rule: P(both aces) = 4/52 × 3/51 = 12/2652 ≈ 0.45%."

# Analytical
p_first_ace = 4/52
p_second_ace_given_first = 3/51
p_both_aces = p_first_ace * p_second_ace_given_first  # 12/2652 ≈ 0.0045

# With replacement (independent):
p_both_aces_with_replacement = (4/52) * (4/52)  # 16/2704 ≈ 0.0059

Q5: "You test 1000 people for a rare disease (1% prevalence). Test is 99% accurate. If someone tests positive, are they likely to have the disease?"

"This is the base rate fallacy — the answer is surprisingly no. Let's compute it. Of 1000 people: 10 have the disease, 990 don't. True positives: 10 × 0.99 = 9.9 ≈ 10. False positives: 990 × 0.01 = 9.9 ≈ 10. Total positives: ~20. Of those, only ~10 actually have the disease: P(disease | positive) ≈ 10/20 = 50%. Despite a 99% accurate test, only half the positive tests are true positives — because the disease is so rare. This is why population screening for rare diseases requires high specificity or confirmatory testing."

Interview Answer

"For conditional probability, my go-to explanation: P(A|B) is the probability of A once you've restricted your view to the world where B is true. The formula P(A|B) = P(A,B)/P(B) is just 'zoom in on the B outcomes, then count what fraction are also A'. The three key things interviewers check: can you apply the complement trick, can you spot when events are or aren't independent, and do you understand the base rate fallacy — that a positive test on a rare disease may still be more likely a false positive than a true positive."