Data science interviews at top tech companies like Paypal, Google, Meta (Facebook), and Samsung are known for their challenging and insightful questions. These questions are designed to evaluate not only your technical expertise but also your problem-solving approach, business acumen, and ability to communicate complex ideas clearly. In this article, we will dive deep into five real-world data scientist interview questions from these companies, thoroughly explaining the concepts, solutions, and thought processes behind each. 

5 Data Scientist Interview Questions from Paypal and Google

1. Paypal: Comparing Expected Values in Uniform Sampling and Multiplication

Question:

What has the larger expected value: sampling a number between 1 and n from a uniform distribution and multiplying it by itself, or sampling two numbers and multiplying them?

Concepts Involved:

• Uniform Distribution
• Expected Value Calculation
• Properties of Independence in Sampling

Detailed Explanation and Solution:

Let’s define the two scenarios:

• Scenario 1: Sample a single integer \( X \) uniformly at random from \( \{1, 2, ..., n\} \), and compute \( X^2 \).
• Scenario 2: Independently sample two integers \( X \) and \( Y \) from \( \{1, 2, ..., n\} \), and compute \( X \cdot Y \).

We are to compare \( E[X^2] \) and \( E[X \cdot Y] \).

Step 1: Compute \( E[X^2] \)

For a discrete uniform distribution over integers \( 1 \) to \( n \):

\[ E[X^2] = \frac{1}{n} \sum_{k=1}^n k^2 = \frac{1}{n} \cdot \frac{n(n+1)(2n+1)}{6} \]

Step 2: Compute \( E[X \cdot Y] \)

Since \( X \) and \( Y \) are independent and identically distributed:

\[ E[X \cdot Y] = E[X] \cdot E[Y] = (E[X])^2 \]

For a discrete uniform distribution:

\[ E[X] = \frac{1}{n} \sum_{k=1}^n k = \frac{n+1}{2} \]

So, \[ E[X \cdot Y] = \left( \frac{n+1}{2} \right)^2 \]

Step 3: Comparison

Let's compare \( E[X^2] \) and \( E[X \cdot Y] \):

- \( E[X^2] = \frac{(n+1)(2n+1)}{6} \) - \( E[X \cdot Y] = \left( \frac{n+1}{2} \right)^2 = \frac{(n+1)^2}{4} \)

Which is larger? Let's compute the ratio:

\[ \text{Ratio} = \frac{E[X^2]}{E[X \cdot Y]} = \frac{(n+1)(2n+1)/6}{(n+1)^2/4} = \frac{(2n+1)}{6(n+1)} \times 4 = \frac{4(2n+1)}{6(n+1)} = \frac{2(2n+1)}{3(n+1)} \]

- For small \( n \) (e.g., \( n = 1 \)), \( E[X^2] = 1, E[X \cdot Y] = 1 \). - As \( n \) increases, let's see the behavior as \( n \to \infty \):

\[ E[X^2] \approx \frac{2n^2}{6} = \frac{n^2}{3} \] \[ E[X \cdot Y] \approx \frac{n^2}{4} \]

Thus, \[ \frac{n^2/3}{n^2/4} = \frac{4}{3} > 1 \]

Conclusion

Sampling one number and squaring it has a larger expected value than sampling two numbers and multiplying them.

2. Meta (Facebook): Estimating User Birthdays Without Direct Input

Question:

We at Facebook would like to develop a way to estimate the month and day of people's birthdays, regardless of whether people give us that information directly. What methods would you propose, and data would you use, to help with that task?

Concepts Involved:

• Feature Engineering
• Predictive Modeling
• Supervised Learning
• Privacy Concerns and Data Sensitivity

Approach and Explanation:

1. Understanding the Problem

The challenge is to estimate a user's birthday (month and day) without them explicitly providing it. The solution must leverage indirect signals available on the platform.

2. Identifying Useful Data Sources

• Friend Activity: Friends posting on a user's wall ("Happy Birthday!") or sending messages.
• User's Own Activity: Increased posts, photos, or status updates on their birthday or in the days leading up to it.
• Indirect Hints: Participation in birthday-related events, "Today is my birthday" mentions, or special profile updates.
• Aggregated Date Patterns: Clusters of similar activity across users whose birthdays are known.

3. Feature Engineering

• Birthday-related Keywords: Count of posts/messages containing "birthday," "happy birthday," etc. on different days.
• Message/Wall Post Volume: Sudden spikes in incoming/outgoing messages.
• Photo Uploads: Increased uploads or tagged photos on a particular day.
• Event Participation: Attending/hosting events with "birthday" in the title.

4. Modeling Approach

• Supervised Learning: Use users with known birthdays as the training set. Train a classifier or regression model to predict the birthday based on engineered features.
• Probabilistic Modeling: For each day, compute the probability that it is the user's birthday given observed features.
• Ensemble Methods: Combine multiple weak signals to improve prediction accuracy.

5. Privacy Considerations

• Ensure all data usage complies with privacy policies and user consent.

Conclusion

By leveraging activity patterns, keyword spotting, and supervised modeling, Facebook can estimate users' birthdays with reasonable accuracy, even without explicit input.

3. Meta (Facebook): Analyzing a 10% Increase in Likes Year-over-Year

Question:

Facebook sees that likes are up 10% year over year, why could this be?

Concepts Involved:

• Metric Analysis
• Root Cause Analysis
• Business Intelligence
• Data Exploration

Analytical Approach and Explanation:

1. Decompose the Metric

• Likes Growth: The total number of likes is a function of the number of active users and likes per user.
• Equation:
  \[ \text{Total Likes} = \text{Active Users} \times \text{Likes per User} \]

2. Potential Causes for Increase

• Increase in Active Users: If the platform has gained more active users, total likes could rise even if likes per user remain constant.
• Increase in Likes per User: Users may be liking more posts on average due to:
  • Improved content recommendation algorithms
  • Changes in UI/UX (e.g., easier to like, double-tap features)
  • Social trends encouraging more engagement
• Platform Changes:
  • Introduction of new features (e.g., reactions, new like buttons)
  • Changes in the newsfeed algorithm surfacing more engaging or likeable content
• External Factors:
  • Events or campaigns driving more engagement (e.g., elections, global events)
  • Seasonal effects (holidays, school vacations)
• Data Artifacts:
  • Changes in how likes are counted or tracked
  • Removal of fake accounts (could cause increase or decrease depending on timing)

3. Investigation Steps

• Break down likes by user segments (age, geography, device)
• Analyze likes per user and user growth separately
• Correlate spikes with product releases or events
• Check for anomalies in tracking or data collection

Conclusion

A 10% increase in likes could be attributed to user base growth, higher engagement per user, product changes, external events, or data artifacts. A thorough decomposition and segment analysis is essential to identify the root cause.

4. Samsung: Probability of Drawing Two Cards with the Same Suit

Question:

What is the probability of drawing two cards (from the same deck of cards) that have the same suit?

Concepts Involved:

• Basic Probability
• Combinatorics
• Card Deck Properties

Step-by-Step Solution:

1. Understanding the Problem

A standard deck has 52 cards, 4 suits (hearts, diamonds, clubs, spades), 13 cards per suit. Draw two cards at random, without replacement.

2. Compute Total Number of 2-Card Combinations

Total ways to draw 2 cards: \[ C(52, 2) = \frac{52 \times 51}{2} = 1326 \]

3. Compute Number of Favorable Outcomes (Same Suit)

• For each suit, number of ways to choose 2 cards from 13: \[ C(13, 2) = \frac{13 \times 12}{2} = 78 \]
• There are 4 suits, so total favorable ways: \[ 4 \times 78 = 312 \]

4. Compute Probability

\[ P(\text{same suit}) = \frac{312}{1326} = \frac{156}{663} \approx 0.235 \]

Conclusion

The probability of drawing two cards with the same suit from a standard deck is approximately 23.5%.

5. Meta (Facebook): Assessing the Impact of Doubling Ads in Newsfeed

Question:

If a PM says that they want to double the number of ads in Newsfeed, how would you figure out if this is a good idea or not?

Concepts Involved:

• Experimentation and A/B Testing
• Key Performance Indicators (KPIs)
• User Experience Measurement
• Revenue Analysis

Analytical Framework and Steps:

Step 1: Define Success Metrics

• Business Metrics: Ad revenue, click-through rate (CTR), cost per mille (CPM)
• User Engagement Metrics: Session time, return rate, number of posts viewed
• User Satisfaction Metrics: Survey scores, complaints, app store ratings

Step 2: Design an Experiment (A/B Test)

• Control Group: Sees current number of ads.
• Treatment Group: Sees double the ads in Newsfeed.
• Randomly assign users to each group to control for confounding variables.

Step 3: Monitor Short-term and Long-term Effects

• Short-term: Immediate changes in ad revenue, engagement, and satisfaction.
• Long-term: Changes in user retention, churn, and brand perception.

Step 4: Analyze Results

• Compare KPIs between groups using statistical tests (t-tests, chi-square tests).
• Segment analysis to see if particular user groups are more/less affected.
• Check for unintended consequences (ad blindness, increased ad blocking, user drop-off).

Step 5: Make a Data-informed Recommendation

• If increased ads significantly boost revenue with minimal user impact, consider rollout.
• If increased ads significantly boost revenue with minimal user impact, consider rollout.
• If user engagement or satisfaction drops, weigh the tradeoffs—short-term revenue vs. long-term user value.
• Consider alternative strategies (e.g., better targeting, ad quality improvements) to balance monetization with user experience.

Step 6: Additional Considerations

• Externalities: More ads might lead to ad fatigue, reducing ad effectiveness over time.
• Advertiser Perspective: Assess impact on advertiser ROI and CPMs.
• Competitor Benchmarking: Analyze industry norms for ad load per user/session.
• User Feedback: Collect qualitative data through surveys and support tickets.

Conclusion

Deciding whether to double ads requires a careful, experimental approach balancing monetization and user experience. Use A/B tests, monitor comprehensive KPIs, and consider both short and long-term impacts before making a recommendation.

6. Google: Code to Generate IID Draws from Distribution X Using Only a Random Number Generator

Question:

Write code to generate IID draws from distribution X when we only have access to a random number generator (Google).

Concepts Involved:

• Inverse Transform Sampling
• Random Number Generators (RNG)
• Probability Distributions

Detailed Explanation and Example

Suppose we have a function rand() that returns a random float uniformly distributed in (0, 1). Given a target distribution X, how do we sample from it?

1. Inverse Transform Sampling Method

Let F be the cumulative distribution function (CDF) of X. The inverse transform sampling method is as follows:

• Generate a uniform random variable \( U \sim \text{Uniform}(0, 1) \).
• Set \( X = F^{-1}(U) \).

This works because the distribution of F^-1(U) matches the target distribution X.

2. Example: Sampling from an Exponential Distribution

Let’s say X is an exponential distribution with rate parameter λ. The CDF is:

\[ F(x) = 1 - e^{-\lambda x} \]

To invert:

\[ u = 1 - e^{-\lambda x} \implies e^{-\lambda x} = 1-u \implies x = -\frac{1}{\lambda} \ln(1-u) \]

Since 1-u is also uniformly distributed, we can write: \[ x = -\frac{1}{\lambda} \ln(u) \] with \( u \sim \text{Uniform}(0, 1) \).

3. Python Implementation

import random
import math

def sample_exponential(lam):
    u = random.random()
    return -math.log(u) / lam

# Generate 5 IID draws from Exp(λ=2)
samples = [sample_exponential(2) for _ in range(5)]
print(samples)

4. General Approach for Arbitrary Distributions

If the inverse CDF is not available in closed form, consider:

• Numerically approximating the inverse CDF (using interpolation or lookup tables)
• Using rejection sampling or acceptance-rejection methods
• Using specialized algorithms for discrete distributions (e.g., alias method for categorical variables)

5. Example: Sampling from a Discrete Distribution

Suppose we want to sample from a categorical distribution with probabilities [0.1, 0.3, 0.6]. We can use the CDF and compare a uniform draw:

import random

def sample_categorical(probs):
    cdf = []
    cumulative = 0
    for p in probs:
        cumulative += p
        cdf.append(cumulative)
    u = random.random()
    for i, threshold in enumerate(cdf):
        if u < threshold:
            return i
    return len(probs) - 1

# Sample from [0.1, 0.3, 0.6]
draws = [sample_categorical([0.1, 0.3, 0.6]) for _ in range(10)]
print(draws)

Conclusion

To generate IID samples from an arbitrary distribution X using only a random number generator, use the inverse transform sampling method (if the inverse CDF is available), or alternative methods like rejection sampling or the alias method for discrete distributions.

Summary Table: Interview Questions, Concepts, and Approaches

Conclusion

Data scientist interviews at leading tech companies demand a deep understanding of probability, statistics, modeling, experimentation, and coding. Mastering questions like those from Paypal, Google, Meta, and Samsung not only prepares you for interviews but hones your analytical thinking and problem-solving skills. Focus on understanding core principles, clearly explaining your reasoning, and always consider both technical and business implications in your answers. Good luck preparing for your next big interview!

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