# probability interview questions and answers pdf

We cant lie -Â Data Science Interviews are TOUGH. The probability and statistics questions that top tech companies and hedge funds ask candidates during the Data Science & Quant Interview process are particularly challenging.

Therefore, for practice, we created 40 actual probability and statistics interview questions used by organizations such as Facebook, Amazon, Two Sigma, and Bloomberg. In our book, Ace The Data Science Interview, we have solutions to all 40 problems as well as to 161 additional data interview problems on SQL, Machine Learning, and Product/Business Sense.

Additionally, you can practice some of these exact Statistics & Probability questions on DataLemur, a platform I created for SQL & Data Science Interviews.

Each statistics interview question on DataLemur has a discussion forum where you can see how others answered the question, which is what makes it entertaining!

## Statistics & Probability Questions Asked By MAANGs | Google Data Scientist | DataInterview

### If 75 customers fall randomly into three equal-sized databases, all partitions are equally likely. Bob and Ben are two randomly selected customers. What is the probability that Bob and Ben are in the same customer database?

This query enables an interviewer to assess your ability to use statistical knowledge to resolve a problem in a practical business scenario. You might need to use a whiteboard, notepad, or even the note feature on your phone to demonstrate calculations for questions like this. This can help your interviewer better understand your thought process.

For instance, “I’d assign a different number from one to 75 for each student with numbers 1 through 25 in group 1, 26 to 50 in group 2, and 51 to 75 in group 3.” I would then choose a random number to give to Bob and Ben. There are 74 random numbers left after Bob picks one, and 24 of them will place him in the same group as Ben. The probability is 24/79. “.

## General questions

Frequently, interviews start off by asking you a few general questions about yourself and your background. They might also inquire about your interest in the job by asking you these questions. Here are some examples of general interview questions that you might encounter as you submit an application for a data science position:

### Can you explain the Bayesian approach to probability?

Using Bayesian statistics, one can use probability to solve statistical problems. Given that the Bayesian approach has numerous real-world applications in quantitative finance and data science, employers might inquire about it during an interview. Your response should demonstrate your comprehension of this strategy and demonstrate that you are also aware of how the frequentist strategy differs from the Bayesian strategy.

“The Bayesian approach defines probability as the measure of believability one has about how a specific event occurs,” for instance Once you have new information about an event, it uses mathematical tools to assist you in updating your beliefs about random events. After finding new evidence, you can modify your beliefs using Bayesian statistics. It is distinct from frequentist statistics, which only consider the results of repeated experiments. “.

The problem is that if you are unfamiliar with probability distributions, you cannot answer this question. Even worse, there are numerous distinct probability distributions available. Do we therefore need to be familiar with every probability distribution?

The PMF of a binomial distribution can be used to provide an answer to the second query, where there are 100 trials overall, 1 success (just one ad), and 0 stories overall. 04 probability of being an ad.

Given that there are 200 students and 2 exams that will be compared, the following results follow:

Throwing a die is a common illustration of a uniform distribution. We would always have an equal chance of getting any of the sides from a 6-sided die.

Additionally, at least three significant statistical issues that are frequently probed in data science interviews are as follows:

## Probability & Statistics Concepts To Review Before YourÂ DataÂ Science Interview

In order to understand the basics of probability, one must first consider sample spaces, fundamental counting, and combinatorial principles. Even though mastering all aspects of combinatorics is not necessary, it is beneficial to comprehend the fundamentals in order to simplify issues. The “stars and bars” counting technique is a well-known illustration of this.

The other core topic to study is random variables. Understanding basic probability distributions, expectation, variance, and covariance concepts is essential.

Understanding the fundamentals of different probability distributions is crucial for modeling random variables. It is essential to comprehend both discrete and continuous examples, as well as expectations and variances. The uniform and normal distributions are the ones that are most frequently discussed in interviews, but there are many other well-known distributions for specific use cases (Poisson, Binomial, Geometric).

Knowing the fundamentals and how to apply them should usually be sufficient. It never hurts to be able to do the derivations for expectation, variance, or other higher moments, for instance, which distribution would waiting for an event be under?

The foundation of statistical inference is hypothesis testing, which can be divided into a few categories. The Central Limit Theorem is the first, and it’s crucial for analyzing large samples of data. Other essential components of hypothesis testing include type I and type II errors, confidence intervals, p-values, and sampling distributions. Finally, it is worthwhile to consider various proportional tests and other hypothesis tests.

The majority of these ideas are essential to A/B testing, which is frequently broached in job interviews at consumer-tech firms like Facebook, Amazon, and Uber. It’s helpful to comprehend not only the technical specifics but also how A/B testing functions conceptually, what the assumptions are, potential pitfalls, and applications to actual products.

Strong knowledge of probability distributions and hypothesis testing are prerequisites for modeling. We will define modeling as the areas that have a significant statistical intersection with machine learning because it is a broad term. This covers subjects like Bayesian statistics, maximum likelihood estimation, and linear regression. Knowing about modeling and machine learning is crucial for interviews that focus on these topics.

## 50 Statistic and Probability Interview Questions and Answers for Data Scientists

Improve Your Skills and Increase Your Confidence with Mock Interviews from Professionals to Ace Your Next Job Interview!

1. What is the difference between quantitative and qualitative data?

Quantitative data is information that has a numerical value attached to it, such as a count or a range. Qualitative data is typically presented verbally and describes a quality or characteristic. Using terms like “tall” or “short” to describe someone’s height, for instance

2. Describe three different encoding methods that can be used with qualitative data.

Label Encoding, One-Hot Encoding, Binary Encoding

The trade-off between the bias- and variance-related errors in a model is known as the bias-variance trade-off. A highly biased model is too straightforward and doesn’t sufficiently fit the training set of data.

An extremely well-fitting model with high variance is unable to generalize beyond the training set of data. Finding a sweet spot to build a machine learning model that fits onto the training data well enough and can generalize and perform well on test data is called the bias-variance trade-off.

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4. Give examples of machine learning models with high bias

5. Give examples of machine learning models with low bias

Decision Trees, Random Forest, SVM

6. With an example, define sensitivity and specificity in machine learning.

Sensitivity is the percentage of times a machine learning model correctly predicts the presence of a disease in an individual. Sensitivity is the percentage of patients with the disease for whom a positive prediction was made.

Specificity is the percentage of people who received a negative prediction despite not having the disease.

7. What distinguishes a Type I error from a Type II error?

When the null hypothesis is accepted even though it is true, this is a type I error. When the null hypothesis is incorrect but is not rejected, an error of type II takes place.

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8. What presumptions are made when creating a model for linear regression?

• There is a linear relationship between the dependent variable (Y) and the independent variable (X).
• Homoscedasticity
• The independent variables aren’t highly correlated with each other (multicollinearity)
• The residuals follow a normal distribution and are independent of each other.
• 9. Name three different types of validation techniques.

• Train-test split
• LOOCV (Leave One Out Cross Validation)
• K-Means Cross-Validation
• 10. What is multicollinearity, and how does it affect a regression model’s performance?

When a dataset contains two or more independent variables that have a high correlation with one another, this is known as multicollinearity. Multicollinearity can make a regression model more difficult to interpret because it makes it difficult to separate out each variable’s individual effects.

11. What is regularization? Why is it done?

To prevent overfitting, regularization is used to introduce some noise into the models. This is usually done by penalizing models with larger weights. Regularization aims to select the simplest model and maintain a trade-off between the model’s bias and variance because a model with more weights becomes more complex.

12. What does the value of R-squared signify?

The amount of change in the dependent variable (Y) that is explained by the independent variable (X) is indicated by the R-squared value. The R-squared value can range from 0 to 1.

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13. What is the Central Limit Theorem?

According to the Central Limit Theorem, the sample mean’s distribution resembles that of the population as a whole as sample sizes increase. This implies that the sample error will diminish as the sample size rises.

14. What is the five-number summary in statistics?

The first quartile, median (second quartile), third quartile, and maximum are all included in the five-number summary. With the aid of a box plot, it provides us with a generalized impression of the appearance of our variable.

15. Explain the process of bootstrapping.

Bootstrapping is used to sample repeatedly from the sample population when there are few samples of the actual population. Each resample will have a different sample mean, and a sampling distribution will be made using these sample means.

16. List three ways to mitigate overfitting.

• L1 and L2 regularization
• Collect more samples
• Using K-fold cross-validation instead of a regular train-test split
• 17. How do you deal with missing data?

Based on the quantity of missing values and the type of variable, you can handle missing data in a number of different ways:

• Deleting missing values
• Imputing missing values with the mean/median/mode
• Building a machine learning model to predict the missing value based on other values in the dataset
• Replacing missing values with a constant
• 18. What are confounding variables?

A confounding variable is a component that influences both the dependent and independent variables, giving the impression that they are causally related.

For instance, there is a strong link between buying ice cream and forest fires. Ice cream sales are rising while there are more forest fires. This is because the confounding variable between them is heat. Ice cream sales and the risk of forest fires both increase as the temperature rises.

19. What is A/B testing? Explain with an example.

A/B testing is a method for evaluating user experience using a randomized experiment. As an illustration, a business wants to test two variations of its landing page with various backgrounds to determine which version increases conversions. Two versions of the landing page are created for a controlled experiment and shown to various groups of people.

20. Explain three different types of sampling techniques.

• Simple random sampling: The individual is selected from the true population entirely by chance, and every individual has an equal opportunity to get selected.
• Stratified sampling: The population is first divided into multiple strata that share similar characteristics, and each strata is sampled in equal sizes. This is done to ensure equal representation of all sub-groups.
• Systematic sampling: Individuals are selected from the sampling frame at regular intervals. For example, every 10th member is selected from the sampling frame. This is one of the easiest sampling techniques but can introduce bias into the sample population if there is an underlying pattern in the true population.
• 21. A model is considered to be __________ if it performs exceptionally well on the training set but poorly on the test set.

22. Explain the terms confidence interval and confidence level.

The probability that the true population parameter falls within a range of two estimates is known as a confidence interval. As multiple samples are repeatedly taken, the level of confidence (for instance, 95% or 99%) refers to the degree of certainty that the true parameter lies within the confidence interval.

23. What is a p-value?

P-values represent the likelihood that the observed result would occur if the null hypothesis were correct. Based on the developed hypothesis, we can establish a threshold for the p-value below which there is little to no chance that the observed result could have occurred. This gives us enough evidence to reject the null hypothesis.

24. What is standardization? Under which circumstances should data be standardized?.

The process of standardization involves putting various variables on the same scale. The distribution of the variables is made to have a mean of 0 and a standard deviation of 1.

Since values that deviate by 2-3 standard deviations from the mean are straightforward to identify, standardizing data can help us better understand extreme outliers. Before supplying data to machine learning models, standardization is also used as a pre-processing method to ensure that all variables receive the same weighting.

25. Give some examples of data points that follow a normal distribution and describe some of the characteristics of a normal distribution.

• The mean, median, and mode in a normal distribution are very close to each other.
• There is a 50% probability that a value will fall on the left of the normal distribution, and a 50% probability that a value will fall on the right.
• The total area under the curve is 1.
• A population’s height and IQ values, for instance, follow a normal distribution.

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26. When should the mean be used as a gauge of central tendency?

The mean is a good indicator of central tendency when data has a normal distribution. However, skewness in data will also pull the mean with it if it is to the left or right, so it is preferable to use the median as a gauge of central tendency.

27. Explain the difference between ridge and lasso regression.

Ridge regression, also known as L2 regression, penalizes the cost function of the model by adding the sum of the squares of the weights. The cost function of the model receives additional absolute weights via Lasso regression or L1 regression.

As it can reduce feature weights to zero and remove extraneous variables, lasso regression can also be used as a feature selection technique.

28. What is the law of large numbers? Provide an example.

The law of large numbers states that as an experiment is independently repeated many times, the outcome of the study approaches the expected value.

For instance, if we toss a coin 1000 times, the likelihood of receiving heads is more likely to be 0. 5 than if we tossed the coin 100 times.

29. How can outliers be identified?

Finding the first and third quartiles will help you spot outliers. Any value lesser than Q1â—â1. 5 X IQR or Q3 + 1. 5 X 1QR is considered an outlier.

30. When assessing a model, when is it preferable to use an F1 score rather than accuracy?

The precision and recall of a model are harmonically represented by the F1 score. The F1 score offers us a more accurate indicator of model performance when dealing with unbalanced datasets than accuracy.

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31. What is selection bias?

When randomization isn’t fully achieved when obtaining a sample population, selection bias is a bias that is introduced. As a result of leaving out a portion of the true population, the sample population won’t accurately reflect the true population.

32. What are some ways to overcome an imbalanced dataset?

An unbalanced dataset is one in which some classes are overrepresented and others are underrepresented. When this occurs, machine learning models frequently make incorrect predictions because they frequently predict a majority class. An unbalanced dataset can be corrected by oversampling or undersampling. By adding data from the minority class to the training dataset at random, oversampling increases the representation of minority class samples.

The opposite is undersampling, where the majority of class samples are chosen and eliminated at random to produce a more evenly distributed distribution of all classes in the dataset.

33. When should you use a t-test and a z-test?

When the population variance is known or when the population variance is unknown but the sample size is substantial, a z-test is employed.

When the population variance is unknown and the sample size is small, a t-test is utilized.

34. What is the difference between a homoscedastic and heteroscedastic model?.

A model is homoscedastic if the variance of the dependent variables is constant across all of the data points. All data points will be spaced from the regression line similarly.

Contrarily, a model is said to be heteroscedastic if the variance varies for each and every data point.

35. Explain why decision trees generally outperform random forests in terms of making predictions.

Random forests typically perform better than decision trees. This is due to the fact that it gets around many of decision trees’ drawbacks. Large decision trees usually lead to overfitting. By only choosing a small subset of variables at each split, random forests address this problem and improve the generalizability of the model created.

A single prediction is made using the results of several weak learners, which is typically more accurate than using the results of a single large decision tree.

36. How can overfitting be detected when building a prediction model?.

Using K-Fold cross-validation for training and testing, overfitting can be identified.

37. If so, why is it that an inaccurate classification model can be highly accurate?

Inaccurate classification models, such as those that only predict the majority class, can be very accurate. For instance, a model that predicts negative on all data points will have a 90% accuracy, which is exceptionally high, if 90% of the samples in the dataset tested negative for disease and 10% tested positive.

However, model performance is still subpar, and this time around, accuracy isn’t a good indicator of how good the model is.

38. There is a right-skewed distribution with a median of 70. What can we conclude about the mean of this variable?.

Given that the positive skew will drag the mean along with it, the mean of this variable will be greater than 70.

39. What is a correlation coefficient?

A correlation coefficient serves as a measure of how closely two variables are related. A coefficient of 0 denotes no correlation, a coefficient of near -1 denotes a strong negative correlation, and a coefficient near +1 denotes a strong positive correlation.

40. How does the performance of the model change when an independent variable has a high cardinality?

The number of categories a single categorical variable has is referred to as its cardinality. A variable with a high cardinality has a wide range of types associated with it.

This can negatively impact model performance if not correctly encoded.

41. What are the different types of selection bias? Explain them.

• Sampling bias: This is the bias introduced by non-random sampling. For example, if you were to survey all university students to understand their view on gender disparity but only surveyed female students. This would introduce a bias into the model and wouldn’t provide a complete picture of the true population.
• Confirmation bias: This is a bias caused by the tendency of an individual to favor information that validates their own beliefs.
• Time-interval bias: This is bias caused by selecting observations that only cover a specific range of time. This can skew the samples collected because it limits the data collected to a specific set of circumstances.
• 42. What presumptions underlie the construction of a logistic regression model?

• Absence of outliers that can strongly impact the model
• Absence of multicollinearity
• There should be no relationship between the residuals and the variable.
• 43. Explain the bagging technique in random forest models.

Bagging stands for bootstrap aggregation. The dataset is sampled repeatedly in random forests using the replacement bootstrap method. Each sample’s different features are split up into separate training sessions for weak learners at each node. The user is given the average or majority class prediction as a final output.

44. Describe three error metrics for a linear regression model.

The MSE, RMSE, and MAE are the three error metrics that are most frequently used to assess performance.

• MSE: The mean squared error measures the average squared difference between true and predicted values across all data points.
• RMSE: The RMSE (root mean squared error) takes the square root of the mean squared error.
• MAE: The mean absolute error takes the average absolute difference between the true and predicted values across all data points.
• 45. Explain the impact of seasonality on a time-series model.

Seasonality is a factor that can affect a time-series model’s performance when developing one. These are cycles that occur repeatedly over a period of time, and the model that is being developed must take these cycles into account. Otherwise, there is a risk of making inaccurate predictions.

Consider developing a model to forecast the volume of hoodies that will be sold in the upcoming months. You won’t take into account seasonal variations in purchasing patterns if you only use data from the beginning of the year to make the prediction. Due to the warmer weather, people would purchase fewer hoodies in March and April than they did in February, a factor that the machine learning model does not take into account.

46. What is the default threshold value in logistic regression models? .

The default threshold value in logistic regression models is 0.5.

47. What situation might call for you to control the threshold value? Is it possible to change this threshold?

It is possible to alter a logistic regression model’s threshold. For instance, when we want to increase the number of true positives and create a model with high sensitivity, we can lower the threshold value to something just below 0. 5 can also be considered a positive prediction.

48. Differentiate between univariate and bivariate analysis.

The univariate analysis only summarizes one variable at a time. Examples of univariate analysis include measures of variance and central tendency. Bivariate analysis examines the connection between two variables, such as covariance or correlation.

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49. Write some Python or R code to determine the correlation between the two variables Y1 and Y2 given.

# Python:

# R:

50. If there is enough evidence to disprove the null hypothesis at a 5% level of significance, then there is also enough evidence to disprove it at a 1% level of significance. ” Is this statement true or false, and explain why.

This statement is sometimes true. We are unable to predict whether the null hypothesis will be rejected because we are unsure of the precise p-value. If the p-value was smaller than 0. At a 1% significance level, there is enough data to rule out the null hypothesis in case 1. However, if the p-value was around 0. 04, the null hypothesis could not be disproved at a 1% significance level.

## FAQ

What are some examples of probability questions?

What is the likelihood of getting exactly two heads when two coins are tossed simultaneously? What is the likelihood of getting a king when a deck of 52 cards is well-shuffled? What is the likelihood of getting exactly five red balls and seven black balls in a bag? What is the probability of getting a black ball?.

How do you solve probability questions?

Adding the probabilities together will yield the probability of a simple event occurring. Your overall odds of winning something, for instance, are 10% + 25% = 35% if your chances of winning \$10 and \$20, respectively, are 10% and 25%, respectively.

What is p-value in interview?

P-value measures how confident we are about the assumption given the observed data, according to a brief definition. Key Values: P-value ranges from 0 to 1. Based on the observed data, a high p-value indicates that we are very confident that the assumption is true.

How do you prepare for a probability and statistics interview?

Probability & Statistics Concepts To Review Before Your Data Science Interview
1. Probability Basics and Random Variables. In order to understand the basics of probability, one must first consider sample spaces, fundamental counting, and combinatorial principles.
2. Probability Distributions. …
3. Hypothesis Testing. …
4. Modeling.