PayPal·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jul 2026

Summary

PayPal data scientist experimentation round had a stats check that felt more like a quiz than a conversation. Quick binomial probability stuff, nothing that required much back-and-forth.

Questions Asked (1)

A fair coin is flipped 10 times. Calculate the total number of possible outcome sequences, the probability of exactly 3 heads, and the probability of at most 3 heads.

A/B Testing & ExperimentationProduct Analytics & Metrics

Author's notes

Multi-part but not actually hard once you remember it's just binomial.

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AI HintsAI Generated

Suggested Approach

Break the problem into three parts: total sequences, exactly 3 heads, and at most 3 heads. Use the binomial coefficient to count sequences and the binomial probability formula, then sum probabilities for the cumulative case. Clearly state assumptions (fair coin, independent flips) and show calculations.

Pro tip: Connect the calculation to A/B testing by noting that this is a binomial experiment, and mention that in practice you'd use a binomial test or confidence interval to compare conversion rates. This shows you can translate a textbook problem into a real-world data science context.

1. Identify the distribution and parameters

Recognize that flipping a fair coin 10 times is a binomial experiment with n=10 and p=0.5. State that each flip is independent and has two outcomes.

2. Calculate total number of sequences

Since each flip has 2 outcomes, the total number of sequences is 2^10 = 1024. This is the sample space size.

3. Compute probability of exactly 3 heads

Use the binomial formula: P(X=3) = C(10,3) * (0.5)^3 * (0.5)^7 = 120 / 1024 ≈ 0.1172. Show the combination calculation.

4. Compute probability of at most 3 heads

Sum probabilities for X=0,1,2,3: P(X≤3) = Σ_{k=0}^{3} C(10,k) / 1024 = (1+10+45+120)/1024 = 176/1024 ≈ 0.1719.

5. Interpret and relate to role

Explain that this cumulative probability is like a p-value for a one-sided test, and discuss how such calculations inform decisions in A/B testing and product analytics.

Key Points to Mention

Binomial distribution assumptions: independent trials, fixed number of trials, constant probability of success, two outcomes.
Binomial coefficient formula: C(n,k) = n! / (k!(n-k)!).
Probability mass function: P(X=k) = C(n,k) * p^k * (1-p)^(n-k).
Cumulative probability: sum of individual probabilities for k=0 to 3.
Connection to A/B testing: calculating probabilities of observed outcomes under null hypothesis, p-values, and statistical significance.
Use of complement or symmetry to simplify calculations (e.g., P(X≤3) = 1 - P(X≥4) but not simpler here).

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