We cant lie -Â Data Science Interviews are TOUGH. Top tech companies ask very tough questions about probability and statistics
That’s why we put together 40 real chances There are answers to all 40 problems in our book, Ace The Data Science Interview. There are also answers to 161 other problems on SQL, Machine Learning, and Product/Business Sense. You can also practice some of these same exact questions on DataLemurs statistics interview questions section.
Probability theory is a vital mathematical concept that underpins data science statistics machine learning, and many other technical fields. With the massive demand for data scientists and analysts today, candidates are frequently tested on their grasp of probabilistic reasoning during the interview process at top tech firms and quant finance companies.
Handling probability theory questions with confidence requires conceptual clarity coupled with problem-solving skills. In this comprehensive guide, we provide tips, detailed explanations, and practice questions to help you ace probability interviews
Why Probability Theory Matters for Data Science Interviews
Data science interviews, especially at leading technology and quantitative finance firms, rigorously assess your foundational knowledge. Probability theory and statistics form a crucial component of these technical core concepts.
Interviewers often come up with a number of questions, ranging from easy probability questions to hard tests of statistical reasoning. They want to see how you use probabilistic methods to get useful information from datasets when there is uncertainty.
Probability theory provides the basic building blocks for statistical data analysis. Understanding probability facilitates inferring properties of populations from random samples. You will also learn how to do predictive modeling and simulation, which are important skills for data science jobs.
Besides the fundamentals, you need a good grasp of probability distributions, hypothesis testing, Bayesian methods, and sampling techniques. Our guide will help you master these through targeted preparation.
Key Probability Theory Concepts to Brush Up
Before an interview, thoroughly review these fundamental areas –
Basic Probability
- Classical and empirical probability
- Conditional probability
- Multiplication rule
- Independence vs. mutual exclusivity
Probability Distributions
- Discrete vs. continuous distributions
- Binomial, normal, Poisson, exponential distributions
- Mean, variance, mode
- Normal distribution and central limit theorem
Bayesian Probability
- Prior and posterior probabilities
- Bayes’ theorem
Sampling & Estimation
- Simple random sampling
- Law of large numbers
- Central limit theorem
Hypothesis Testing
- Null and alternative hypotheses
- p-values
- Type I and Type II errors
- t-tests, z-tests, ANOVA
Armed with a solid grasp of these topics, you can approach probability questions with confidence. Let’s now see some example questions and strategies to solve them effectively.
Probability Theory Interview Questions and Detailed Explanations
Probability theory questions test your conceptual understanding and your ability to apply probabilistic techniques to data problems. Here we cover examples ranging from basic probability puzzles to more advanced Bayesian reasoning.
Q1. You roll two six-sided dice. What is the probability that the sum is even?
To solve this, first list out the possible outcomes and identify those with an even sum –
(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)
Out of 36 total possible outcomes, 18 have an even sum.
Therefore, the required probability is 18/36 = 1/2.
Q2. An urn contains 3 red balls and 5 black balls. Two balls are drawn randomly without replacement. What is the probability both balls are red?
- Total balls = 3 + 5 = 8
- Possible balls in first draw = 8
- If first ball is red, balls for second draw = 7
- Remaining red balls for second draw = 2
- Required probability = (3/8) x (2/7) = 6/56 = 3/28.
Q3. A college has 56% female students. If we select 35 students randomly, what is the probability that at least 20 are female?
Let X = number of female students in a sample of 35
X follows the binomial distribution with n = 35, p = 0.56.
We need to calculate P(X ≥ 20) using the binomial CDF.
This gives a probability of 0.697 or 69.7%
Q4. Suppose the time between arrivals of customers at a store follows an exponential distribution with a mean of 10 minutes. What is the probability that the next customer arrives within 5 minutes?
For an exponential distribution with rate parameter λ, the probability that a random variable X is less than x is given by:
P(X < x) = 1 – e^(-λx)
Here λ = 1/mean = 1/10 per minute
x = 5 minutes
Therefore, required probability = 1 – e^(-1/10 * 5) = 0.3935 or 39.35%
Q5. Let X and Y be independent random variables with standard normal distributions. What is the distribution of Z = X + Y?
For independent normal random variables X and Y with mean 0 and variance 1, the sum Z = X + Y is also normally distributed per the central limit theorem.
Its mean is μX + μY = 0 + 0 = 0
Its variance is σX^2 + σY^2 = 1 + 1 = 2
Therefore, Z ~ N(0, 2), i.e. normal distribution with mean 0 and variance 2.
Q6. You have an urn with 3 red balls and 5 blue balls. You draw a ball, record its color, and put it back in the urn. This is repeated 10 times. What is the probability of getting exactly 3 red balls?
Each draw is an independent trial with constant 3/8 probability of drawing red ball.
Let X = number of red balls in 10 trials
X follows a binomial distribution with n = 10, p = 3/8
We want P(X = 3) = (10C3) x (3/8)^3 x (5/8)^7 = 0.218 or 21.8%
Q7. Suppose the number of accidents per day at a factory follows a Poisson distribution with a mean of 3.5. What is the probability that on a randomly selected day there are more than 5 accidents?
Let X = number of accidents per day
X ~ Poisson(3.5)
We want, P(X > 5)
= 1 – P(X ≤ 5) [from Poisson CDF]
= 1 – e^(-3.5) * (3.5^0/0! + 3.5^1/1! + … + 3.5^5/5!)
= 0.147 or 14.7%
Q8. You have a biased coin that has a 60% probability of landing heads. You toss it 120 times. What is the probability that it lands heads between 70 and 80 times?
Let X = number of heads in 120 tosses of the biased coin
X ~ Binomial(120, 0.6)
We want, P(70 ≤ X ≤ 80)
= P(X ≤ 80) – P(X ≤ 69) [Using binomial CDF]
= 0.25
Therefore, required probability is 0.25 or 25%
Tips for Tackling Probability Theory Interview Questions
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Clarify the problem statement and identify the key inputs provided.
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Determine if it involves conditional probability or independence of events.
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Figure out which probability distribution to use based on the problem description. Common options include binomial, normal, Poisson, and exponential distributions.
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Write down the precise formulation in terms of probability or probability density functions. Identify the required probabilities to calculate.
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For problems involving multiple stages, break them down step-by-step into simpler conditional probability calculations.
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Draw a tree diagram or table to organize your workings if dealing with combinations of events.
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For problems involving central limit theorem or law of large numbers, reason about what happens asymptotically as number of trials increases.
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Check your final probability values lie between 0 and 1. Verify solutions if time permits.
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Communicate your approach clearly to the interviewer when explaining your solution.
Developing fluency in translating real-world situations into probability formulations takes practice. Work through examples from statistics textbooks and previous interview prep guides. Mastering probability theory will stand you in good stead for acing technical interviews and excelling in data science roles.
Solutions To Probability Interview Questions
Problem #1 Solution:
We can use Bayes Theorem here. Let’s call the situation where we flip an unfair coin U and the situation where we flip a fair coin F. Since the coin is chosen randomly, we know that P(U) = P(F) = 0. 5. Let 5T denote the event where we flip 5 heads in a row. Then we are interested in solving for P(U|5T), i. e. , the chance that we are flipping an unfair coin, since we have seen five tails in a row
We know P(5T|U) = 1 since by definition the unfair coin will always result in tails. Additionally, we know that P(5T|F) = 1/2^5 = 1/32 by definition of a fair coin. By Bayes Theorem we have:
[P(U|5T) = frac{P(5T|U) * P(U)}{P(5T|U) * P(U) + P(5T|F) * P(F)} = frac{0. 5}{0. 5 + 0. 5 * 1/32} = 0. 97].
Therefore the probability we picked the unfair coin is about 97%.
Problem #5 Solution:
By definition, a chord is a line segment whereby the two endpoints lie on the circle. Therefore, two arbitrary chords can always be represented by any four points chosen on the circle. If you choose to represent the first chord by two of the four points then you have:
[{4choose2} = 6 ]
pick the two points that will show chord 1 (and the other two points that will show chord 2). However, keep in mind that we are counting each chord twice because a chord with endpoints p1 and p2 is the same as a chord with endpoints p2 and p1. Therefore the proper number of valid chords is:
Among these three configurations, only exactly one of the chords will intersect, hence the desired probability is:
Problem #13 Solution:
Let X be the number of coin flips needed until two heads. Then we want to solve for E[X]. Let H denote a flip that resulted in heads, and T denote a flip that resulted in tails. Note that E[X] can be written in terms of E[X|H] and E[X|T], i. e. the expected number of flips needed, conditioned on a flip being either heads or tails respectively.
Conditioning on the first flip, we have:
[E[X] = frac{1}{2}(1+E[X|H]) + frac{1}{2}(1+E[X|T])]
Keep in mind that E[X|T] = E[X] because we have to start over to get two heads in a row if a tail is flipped.
To find E[X|H], we can make it depend on the next result, which could be heads (HH) or tails (HT).
Therefore, we have:
[E[X|H] = frac{1}{2}(1+E[X|HH]) + frac{1}{2}(1+E[X|HT])]
If the outcome is HH, then E[X|HH] = 0 because the goal was met. If the outcome is HT, then E[X|HT] = E[X] because a tail was flipped, we need to start over.
[E[X|H] = frac{1}{2}(1+0) + frac{1}{2}(1+E[X]) = 1 + frac{1}{2}E[X]]
Plugging this into the original equation yields E[X] = 6 coin flips
Problem #15 Solution:
Consider the first n coins that A flips, versus the n coins that B flips.
There are three possible scenarios:
- A has more heads than B
- A and B have an equal amount of heads
- A has less heads than B
In case 1, A will always win (no matter what coin comes up), and in case 3, A will always lose (no matter what coin comes up). By symmetry, these two scenarios have an equal probability of occurring.
Denote the probability of either scenario as x, and the probability of scenario 2 as y.
We know that 2x + y = 1 since these 3 scenarios are the only possible outcomes. Now letâs consider coin n+1. If the flip results in heads, with probability 0. 5, then A will have won after scenario 2 (which happens with probability y). Therefore, Aâs total chances of winning the game are increased by 0. 5y.
Thus, the probability that A will win the game is:
[x + frac{1}{2}y = x + frac{1}{2}(1-2x) = frac{1}{2}]
Problem #18 Solution:
Let B be the event that all n rolls have a value less than or equal to r. Then we have:
since all n rolls must have a value less than or equal to r. Let A be the event that the largest number is r. We have:
[B_r = B_{r-1} cup A_r]
and since the two events on the right hand side are disjoint, we have:
[P(B_r) = P(B_{r-1}) + P(A_r)]
Therefore, the probability of A is given by:
[P(A_r) = P(B_{r}) – P(B_{r-1}) = frac{r^n}{6^n} – frac{(r-1)^n}{6^n}]
Probability Basics and Random Variables
The beginnings of probability start with thinking about sample spaces, basic counting and combinatorial principles. Even though you don’t need to know everything about combinatorics, it can help you solve problems if you know the basics. One classic example here is the âstars and barsâ counting method.
The other core topic to study is random variables. Knowing concepts related to expectation, variance, covariance, along with the basic probability distributions is crucial.
For modeling random variables, knowing the basics of various probability distributions is essential. Understanding both discrete and continuous examples, combined with expectations and variances, is crucial. Most of the time, interviews talk about the Normal and Uniform distributions. However, there are many other well-known distributions that can be used in different situations, such as the Poisson, Binomial, and Geometric ones.
Most of the time knowing the basics and their applications should suffice. For instance, what kind of distribution would you be in if you flipped a coin? What about if you were waiting for something to happen? It never hurts to know how to figure out the derivations for expectation, variance, or other higher moments.
Hypothesis testing is the backbone behind statistical inference and can be broken down into a couple of topics. The first is the Central Limit Theorem, which plays an important role in studying large samples of data. Other core elements of hypothesis testing: sampling distributions, p-values, confidence intervals, type I and II errors. Lastly, it is worth looking at various tests involving proportions, and other hypothesis tests.
A/B testing, which is often talked about in job interviews at consumer tech companies like Facebook, Amazon, and Uber, is based on most of these ideas. It’s helpful to know not only the technical details of A/B testing, but also how it works in general, what the assumptions are, what the possible problems are, and how it can be used in real-life products.
Modeling relies on a strong understanding of probability distributions and hypothesis testing. Schemas are a broad term, so we’ll just call the areas where statistics and machine learning come together “modeling.” This includes topics such as: linear regression, maximum likelihood estimation, &Â bayesian statistics. For interviews focused on modeling and machine learning, knowing these topics is essential.
Statistics & Probability Interview Questions For Data Science | Data Science Training | Simplilearn
FAQ
What are good questions to ask about probability?
What is a lazy movie raters Netflix probability interview question?
Is probability asked in data science interview?
What is the probability to get two consecutive threes when you roll a dice three times?
How many probability interview questions are there?
In this article, we list 41 probability interview questions and provide examples of the types of probability questions an interviewer might ask you to assess your statistical knowledge. A hiring manager may ask general probability interview questions during an interview to learn more about you and your background.
How many questions are there in probability?
In this article, I will list 12 questions in probability for you to practice. I will list common and classic questions in four topics: general probability, Binomial distribution, conditional probability, and Bayesian probability. I provide my answers to these questions in the back so that you can compare your solutions to mine.
What is probability theory?
Probability theory is a branch of mathematics concerned with the analysis of random phenomena.
What are the most popular probability distributions in a data science interview?
Binomial, uniform, and Gaussian distributions are the most popular ones in a data science interview among all probability distributions. And if you’re really new to probability distribution, you can start with these three before branching out to the other probability distributions.
