These 30 probability and statistics interview questions will help you get ready for your data science job and do well on it.
Questions about statistical or probability concepts in a data science interview can be tricky to handle. This is because unlike a product question, statistics and probability questions have a definite right or wrong answer. This means that your knowledge about specific statistics and probability concepts will be fully tested during the interview. So, before the data science interview, you should review your statistics and make sure you are fully prepared.
We’ll help you improve your statistics and probability skills by giving you thirty real-life interview questions from different companies, along with their answers.
Note that in this article, we’re only going to discuss the interview questions and their solutions. The theoretical concept will only be explained briefly. If you’d like to brush up on your statistical and probability knowledge, you might want to read our complete guide here.
There are at least three big topics in probability that are commonly asked in a data science interview:
We’re going to go through all of these three topics one-by-one. Let’s start with independent and dependent events.
Probability distributions are a key concept in statistics and data science interviews. Understanding how to apply them to real-world problems is essential. In this comprehensive guide, we’ll explore some of the most common probability distribution interview questions you’re likely to encounter and provide example answers to help you prepare.
What Are Probability Distributions?
A probability distribution describes the probabilities associated with a random variable It shows the range of possible values a random variable can take and the likelihood of observing each value
Some common probability distributions include:
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Normal distribution – Models quantitative data that clusters around a mean. Useful for natural processes like human heights.
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Binomial distribution – Models binary outcomes over multiple trials, Useful for modeling success/failure situations like coin flips
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Poisson distribution – Models counts of rare events happening over an interval. Useful for modeling events like website clicks per minute
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Uniform distribution – Models data evenly distributed across a range. Useful for situations with equal probability outcomes like dice rolls.
Frequently Asked Interview Questions
Below we’ll explore some common probability distribution questions that come up in data science interviews:
Q: Explain probability distribution.
A probability distribution describes the probabilities associated with different outcomes for a random variable. It provides a mathematical function that maps outcomes to their likelihood of occurrence. The total area under the probability distribution equals 1, representing all possible outcomes.
Key properties include the distribution shape, center, and spread. The shape indicates the general pattern of probabilities such as symmetric, skewed, uniform. The center represents the central tendency like the mean or median. The spread describes the variability through measures like standard deviation.
Q: How would you generate a random number between one and seven with only a single die?
This can be modeled with a uniform distribution. The outcomes 1 to 6 on a single die are equally likely. To generate a random number from 1 to 7, we can roll the die once and add 1 to the outcome. This results in an evenly distributed random number between 1 and 7, with each value equally likely.
Q: If you draw from a normal distribution with known values of parameters, how do you generate draws in a uniform distribution?
The probability integral transform can be used here. It states that if X is a continuous random variable with cumulative distribution function F(x), then Y=F(X) will have a uniform distribution over the interval (0,1).
So if we have a normal distribution N(μ, σ^2), we can generate uniform draws U ~ uniform(0,1) and then apply the inverse CDF of the normal distribution to get X=F^-1(U) where X will follow the desired normal distribution.
Discrete vs Continuous Distributions
Q: What is the difference between discrete and continuous probability distributions?
Discrete distributions deal with events that can take on only distinct, separate values like counts. Continuous distributions involve events that can take on any value over a range.
Discrete distributions are characterized by probability mass functions (PMF) that provide probabilities for each distinct outcome. Continuous distributions have probability density functions (PDF) that define likelihoods across a continuum.
Examples of discrete distributions include binomial, Poisson, geometric. Continuous distribution examples include normal, uniform, exponential.
Q: When would you use a discrete vs continuous distribution for modeling?
Discrete distributions are used when dealing with discrete or count data. For example, number of customers visiting a store, defect rates in manufacturing, or survival counts in medical testing.
Continuous distributions apply when data can take on any value within a continuous range. Examples include time between events, measurement values like distance or temperature, or metrics like income level.
The nature of the problem and type of data determine whether a discrete or continuous probability distribution would be appropriate for modeling.
Normal Distribution
The normal distribution is one of the most common and useful probability distributions. Several interview questions test your grasp of its properties and applications:
Q: What is the significance of the 68–95–99.7 rule in a normal distribution?
This rule states that in a normal distribution:
- About 68% of values fall within 1 standard deviation of the mean
- About 95% of values fall within 2 standard deviations of the mean
- About 99.7% of values fall within 3 standard deviations of the mean
This provides a shorthand to estimate ranges and probabilities for outcomes based on their standard deviation from the mean. It’s useful for statistical inference tasks like estimating confidence intervals.
Q: How can you test whether a dataset follows a normal distribution?
Visual methods include histograms and Q-Q plots. Statistical tests include the Shapiro-Wilk test, Kolmogorov-Smirnov test, and Anderson-Darling test. These assess the skew, kurtosis, and difference between the empirical distribution and normal distribution.
If the p-values from such tests are above significance levels like 0.05, we fail to reject the null hypothesis that the data is normally distributed.
Q: What are some real-world examples where normal distributions would apply?
Some examples include:
- Measurement errors in scientific experiments
- Human characteristics like height or IQ scores
- Natural processes like radioactive decay rates
- Errors in precise manufacturing processes
- Fluctuations in stock prices
The normal distribution models many natural processes where variability exists around a central tendency. Its symmetric bell shape fits many real-world scenarios.
Poisson Distribution
The Poisson distribution is useful for modeling rates of rare events. Interviewers may ask about interpreting and applying it:
Q: How do you know when to use a Poisson distribution?
Use a Poisson distribution when:
- You’re counting the occurrences of discrete events
- The events happen randomly in time/space with a known constant rate
- The events are independent from each other
It works well for modeling rare events over time/space like number of website clicks per minute or defects on a factory line.
Q: If the Poisson distribution rate (λ) increases, how is the distribution affected?
As the rate λ increases, the Poisson distribution shifts right. The mean increases by λ as well. The variance also increases by λ.
So a higher rate parameter implies more occurrences of events in the same interval, resulting in a right-skewed distribution.
Q: What are some examples of Poisson distribution applications?
Some examples include:
- Modeling demand for emergency services like 911 calls
- Predicting airline baggage claims for lost items
- Estimating traffic accidents at an intersection
- Analyzing rare diseases cases in a population
- Simulating particle detection in physics experiments
The Poisson distribution is well suited for situations involving rare, sporadic events. Its rate parameter provides a useful relationship between the mean and variance.
Bayesian Statistics
Bayesian statistics questions often involve conceptual understanding of priors/posteriors. For example:
Q: Explain prior and posterior distributions in Bayesian statistics.
The prior distribution in Bayesian statistics encodes our beliefs about a parameter before observing data. It acts as a baseline knowledge state.
The posterior distribution incorporates observed data to update and improve our prior belief. It combines the prior and likelihood to provide a data-informed probability distribution for the parameter.
As more data comes in, the posterior becomes the new prior for the next iteration. This allows Bayesian learning as beliefs update.
Q: How would you explain Bayes’ theorem in simple terms?
Bayes’ theorem provides a mathematical rule for combining prior knowledge with observed data to calculate updated beliefs. We can explain it simply as:
Posterior = Likelihood x Prior / Evidence
Where:
- Posterior is our updated belief after seeing data
- Likelihood is the probability of data under a given parameter value
- Prior is our initial belief about parameters before data
- Evidence is a scaling factor
So Bayes’ theorem updates priors into posteriors by incorporating the evidence provided through new data.
Q: What are conjugate priors and why are they useful?
Conjugate priors lead to posterior distributions from the same family as the prior. For example, Beta priors are conjugate to Binomial likelihoods.
Conjugacy is useful because it makes Bayesian analysis mathematically convenient. The same form of the distribution is maintained after updates, making iterative Bayesian inference simpler.
Tips for Acing Probability Distribution Interview Questions
Preparing for probability distribution questions takes practice and experience working with various distributions. Here are some tips:
- Understand key properties like shape, center, spread, PMF/PDF
- Memorize important rules like 68-95-99.7 for normal distributions
- Practice applying distributions to sample problems
- Brush up on Bayesian concepts like priors, posteriors, conjugacy
- Review statistical tests for distribution assumptions
- Know which distributions apply to different real-world scenarios
With a combination of conceptual, practical, and theoretical knowledge, you’ll be equipped to tackle any probability distribution problems an interview throws your way!
Independent and Dependent Events
In probability, an event is said to be independent if the chance of one event happening doesn’t change the chance of another event happening.
The most common example of independent events is throwing two different dice or tossing a coin several times. The chance of getting a tail on the second flip of the coin wouldn’t change based on what happened on the first flip. The probability of us getting a tail will always be 0. 5.
On the other hand, an event is dependent if the chance of one event happening changes the chance of another event happening.
An example of a dependent event is drawing cards from a deck of cards. Say we want to know how likely it is that a deck of cards will show us a red heart. If this is your first time drawing a card, the chance of getting a red heart is 13 out of 52. Let’s say that you got a black spade in the first draw. Then, since you’ve already drawn one card, the chance of getting a red heart on the second draw is 13/51 instead of 13/52.
Here are some examples of data science interview questions from different companies that will test our knowledge of events that depend on and on their own:
“How likely is it that I will draw two cards from the same deck that are the same suit?”
This is an example of a dependent event. The probability that two events will occur in the case of dependent event can be defined as:
That is, the chance that both event A and event B will happen is the same as the chance that event A will happen times the chance that event B will happen based on the outcome of event A.
In our case, there are four suites in a deck of cards, and each suite has 13 cards.
In the first draw, our probability to get a card with a specific suite would be 13/52. The odds of getting a card with the same suit as the first one would drop from 13/52 to 12/51 in the second draw. Hence:
Question from Jane Street:
“What is the probability of choosing 2 queens out of a deck of cards?”
This is also an example of a dependent event. In the first draw, our probability of getting a queen is 4/52. If we get a queen in the first draw, the odds of getting another queen in the second draw are 3/51. Hence:
“Let’s say you have 2 dice. What is the probability of getting at least one 4?”
However, unlike other questions, this one is an example of an independent event, since the result of throwing one die wouldn’t change the result of throwing the second die.
Let’s say that:
A = getting a 4 in the first dieB = getting a 4 in the second die
The probability of independent events A and B both to occur can be defined as:
If you know the odds of getting at least one 4, you can figure out the odds of the union of two events:
We know that the probability of us getting any specific outcome from throwing a die is ⅙. Thus,
“Three ants are sitting at the three corners of an equilateral triangle. Each ant randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?”.
Although it’s implicit, this is the case of an independent event. Each ant can randomly pick the direction, either to the left or to the right. The choice of one ant to go to the left wouldn’t change the choices of the other two ants about whether to go to the left or the right.
Since the decision is random, then the probability of an ant to pick a certain direction is 0. 5. The three ants won’t run into each other if they all go to the left or the right.
Hence:
Permutations and combinations probably sound similar and we have probably used the two words interchangeably in real life. There is a clear difference between the two ideas, though, and it’s important to know the difference between combination and permutation because they have different formulas.
One big difference between permutation and combination is the importance of order. The order is very important in permutation but not in combination. This concept of order will be explained more deeply in the examples of data science interview questions below.
“How to find who cheated on essay writing in a group of 200 students?”
There are different ways on how we can find who’s cheating in an exam. One way to do this is by comparing a pair of student exams one-by-one.
Comparing student A’s test with that of student B is the same thing as comparing student B’s test with that of student A, if we think about it. In other words, A, B = B, A. The order doesn’t matter.
Since the order doesn’t matter, then we can use the concept of combination. The general equation of combination is:
that is, n is the number of items and k is the number of items to be chosen.
Since there are 200 students and there 2 exams that will be compared, then we have:
“From a deck of cards numbered from 1 to 100, we draw two cards at random. What is the chance that a number on one card is exactly twice the number on the other card?”
This question can also be answered with the concept of combination. This is because the order doesn’t matter when we draw two cards from the same deck. This means that getting a 10 in the first draw and a 40 in the second draw is the same thing as getting a 40 in the first draw and a 10 in the second.
Thus, by plugging values that we know from the question into the combination equation we will get:
which means that we have 4950 combination pairs.
Out of those 4950 possible combinations, there are 50 times that one card is the double of the other card. This is because we have 100 cards in total. Thus, we can compute the probability as:
“Three people, and 1st, 2nd and 3rd place at a competition, how many different combinations are there?”
The order is important in this question because being in the first spot is not the same as being in the second or third spot.
Now, let’s say we have athletes A, B, and C in positions 1, 2, and 3. Then A, B, and C are not the same as C, B, and A, nor are B, A, and C. Thus, we’re dealing with the concept of permutation in this question.
The general equation for permutation problem is:
where n is the number of items and k is the number of items that need to be ordered.
In the questions, we have three athletes and three places to be ordered, hence:
A knowledge about probability distribution is a must before you’re going to a data science interview. Question about probability distributions is one, if not, the most popular data science interview question out there.
Below is one interview question that test your general knowledge about probability distributions:
“What is an example of a dataset with a non-Gaussian distribution?”
To answer this question, we can give an example of data with a binomial distribution, like the number of times you’ll get 500 tails when you flip a coin 1000 times, or the number of times you’ll get two 5s when you roll a die 10 times, etc.
You can’t answer this question if you don’t know anything about probability distributions. To make things worse, there are a lot of different probability distributions out there. So do we need to know all of the probability distributions?.
In a data science interview, the probability distributions that come up most often are the binomial, uniform, and Gaussian ones. Plus, if you’re brand new to probability distribution, you can start with these three before moving on to the others.
Probability distribution questions usually come up in a data science interview. You may be asked to find the expected value of a distribution or the probability mass function (PMF) or probability density function (PDF) of a distribution.
Let’s start with binomial distribution.
Binomial distribution is a type of discrete probability distribution that shows how likely it is that something will happen after a certain number of tries.
The probability mass function (PMF) of binomial distribution is as follows:
where n is the number of trials and k is the number of successes. Meanwhile, the expected value of binomial distribution can be computed as follows:
The following are examples of interview questions from different companies that are related to data science and cover the idea of binomial distribution.
Question from Verizon Wireless:
“What is the probability of getting one 5 on throwing dice 7 times?”
This question can be answered by simply plugging in values into the equation of binomial distribution. As we look at one 5, we can say that the number of successes is one and the number of trials is seven. Meanwhile, the probability of getting a 5 in a single throw is, as we all know, ⅙. Hence:
Question from Jane Street:
“Whats the probability of obtaining 2 tails in 5 coin flips?”
Just like the last question, this one can be answered by entering numbers into the PMF equation of the binomial distribution. In this case, we want to get two tails, so the number of successes is two, and the total number of trials is five. The probability of getting a tail in each fair coin toss is 0. 5. Hence:
“A discount coupon is given to N riders. The probability of using a coupon is P. What is the probability that one of the coupons will be used?”.
Putting the numbers into the PMF equation of the binomial distribution is another way to answer this question.
We can figure out from the question that there will be one success (because only one coupon will be used) and that there are N items. The chance of success in a single trial is P.
“A $5 discount coupon is given to N riders. The probability of using a coupon is P. What is the expected cost for the company?”.
In contrast to the last question, we now need to find the expected value of a variable with a binomial distribution instead of the PMF. We can find the answer by putting the numbers into the expected value of binomial distribution equation.
From the equation above, we have N coupons and the probability of using a coupon is P.
Thus, the expected value would be:
And the expected cost would be:
“We have two options for serving ads within Newsfeed: 1. Out of every 25 stories, one will be an ad 2. Every story has a 4% chance of being an ad. For each choice, how many ads do you think will be shown in 100 news stories? If we choose option 2, what is the chance that a user will only see one ad in 100 stories?
This question tests your knowledge on both expected value and the PMF of binomial distribution.
The first question, which is the expected number of ads shown in 100 news stories would be:
For the second question, the PMF of a binomial distribution can be used to give an answer. In this case, there are 100 trials, one success (a single ad), and no stories with 0 04 probability of being an ad.
Uniform distribution can be classified as both discrete and continuous probability distribution, depending on the use case. It finds the chance of an event with n possible outcomes, each of which has an equal chance of happening. Because of this, it has a flat PMF/PDF.
The common example of a uniform distribution is throwing a die. Our probability of getting any of the sides from a 6-sided die would always be ⅙.
The expected value of a discrete uniform distribution is:
where a is the minimum possible outcome and b is the maximum outcome. For instance, if we roll a six-sided die, the worst thing that could happen is 1 and the best thing that could happen is 6.
Below are the examples of data science interview questions that test your knowledge about uniform distribution.
Question from Jane Street:
“What is the expectation of a roll of a die?”
To easily answer this question, we can plug the numbers into the following formula for the expected value of a uniform distribution:
“Suppose you roll a die and earn whatever face you get. Now suppose you have a chance to roll a second die. If you roll, you earn whatever face you get but you forfeit earnings from the first round. When should you roll the second time?”.
This question is somewhat an extension from the previous question. As you already know from the last question, a six-sided die roll is likely to happen when:
To answer this question, we need to think it like this:
If we get more than 3. 5 on the first roll, which is the expected value of a single roll, we shouldn’t roll the second die and should keep the money we made. Meanwhile, if we get less than 3. 5, then we should roll the second die.
“If you pick a number between 1 and N from a uniform distribution and multiply it by itself, or if you pick two numbers from the same uniform distribution and multiply them, which has the higher expected value?”
This question can be interpreted to either one of these two:
The first way to look at it is to take one sample, multiply it by itself, and then figure out what the expected value is.
To answer this question, we need to know the general equation of variance for a variable that has a normal distribution:
- We pick a number from 1 to N for the first case. Let’s call it X. This X can be multiplied by itself to get X^2. Its expected value is E(X^2).
- For the second case, we pick two numbers at random from 1 to N. Once we multiply them together, we get E(X)E(X), which is the expected value for both numbers.
We know that the value of variance should always be positive by looking at the above equation. To fulfill this condition, then E(X^2) has to be larger than E(X)^2. We can see that the expected value is always higher when we pick a number between 1 and N and multiply it by itself.
The second way to look at it is to take one sample, figure out what its expected value is, and then multiply that expected value by itself.
- Pick a number from a uniform distribution between 1 and N for the first case. Then, multiply the expected value of that number by itself to get E(X)^2.
- In the second case, we pick two separate numbers at random and multiply their expected value by 2. This gives us E(X)E(X) = E(X)^2.
Thus, we can conclude that both methods result in similar expected values.
The mean and the standard deviation are the two numbers that describe the Gaussian distribution, which is also called the normal distribution. It looks like a bell curve.
Most of the time, interview questions about normal distribution are asked along with questions about other topics in inferential statistics, like how to figure out p-Value, sample size, margin of error, confidence interval, and hypothesis testing.
You can see the example interview questions of any of these in the following section.
Along with these, there are at least three big questions about statistics that are often asked in data science interviews.
- Measure of center and spreads (mean, variance, standard deviation)
- Inferential statistics
- Bayes’ theorem
Let’s discuss the measure of center and spread first.
Also, check out our Comprehensive Statistics Cheat Sheet to learn about important probability and statistics terms and equations.
