Probability & Bayes' Theorem

Random Variables & Distributions

How do you define and use random variables in probability?
Junior
What is the difference between discrete and continuous probability distributions?
Junior
What is a probability density function (PDF)?
Junior
What is the role of the cumulative distribution function (CDF)?
Junior
How would you explain a probability distribution to a layperson?
Junior
What is the difference between a PDF and a CDF, and how are they related?
Junior
How do Probability Mass Functions (PMF) and Probability Density Functions (PDF) differ?
Mid
What are probability distributions and their importance in machine learning?
Mid
How could a probability distribution be non-normal, and can you provide an example scenario such as a uniform distribution?
Mid
How do you find the distribution of a function of a random variable?
Senior

Probability Fundamentals

What are independent and dependent events in probability?
Junior
What is probability, and how is it used in machine learning?
Junior
Can you define the terms 'sample space' and 'event' in probability?
Junior
What are the basic rules of probability, including the addition rule, multiplication rule, and complement rule?
Junior
What are the Kolmogorov axioms of probability?
Junior
What is the difference between mutually exclusive events and independent events?
Junior
What is the Law of Large Numbers?
Mid
Can you explain the inclusion-exclusion principle for the probability of a union of events?
Mid
What is the difference between the Law of Large Numbers and the Central Limit Theorem?
Mid

Conditional Probability & Independence

How would you calculate the probability of 'at least one' event occurring using the complement rule?
Junior
Can you explain the differences between joint, marginal, and conditional probabilities?
Mid
Explain why P(A|B) is not necessarily equal to P(B|A).
Mid
What is the law of total probability and how do you use it to partition a sample space?
Mid
Can you explain the chain rule (general multiplication rule) for probability?
Mid
How can assuming independence in probabilistic models lead to inaccuracies?
Senior
What is the difference between independence and conditional independence?
Senior

Common Distributions

What are the characteristics of a normal distribution?
Junior
What are z-scores and standardization, and what does the 68-95-99.7 rule tell you about a normal distribution?
Junior
When would you model a scenario with a uniform distribution, and what are its mean and variance?
Junior
When would you use a Binomial distribution, and what are its key properties such as mean and variance?
Mid
Explain the utility of the Binomial distribution in machine learning.
Mid
How does the Poisson distribution differ from the Binomial distribution?
Mid
What is the relevance of the Bernoulli distribution in machine learning?
Mid
When do you use the binomial, Poisson, and normal distributions?
Mid
How would you compute the expected number of trials until the first success?
Mid
When does the geometric distribution arise, and what are its mean and variance?
Mid
Can you explain the Central Limit Theorem and its significance in machine learning?
Senior
What is the memorylessness property of the exponential distribution and what does it mean?
Senior
When can the Poisson distribution be used to approximate the Binomial distribution?
Senior

Expectation Variance & Moments

What is expected value and how is it used?
Junior
Is expected value a value you actually expect to see?
Junior
What is the difference between probability and likelihood?
Mid
Can you define expectation, variance, and covariance?
Mid
Can you explain the linearity of expectation?
Mid
What are the key properties of variance, for example how variance behaves for sums of random variables?
Mid
What is the difference between covariance and correlation?
Mid
How would you decide whether a game or bet is worth playing based on its expected value?
Mid
What is a central moment of a probability distribution of a random variable about its mean, and can you describe the zeroth, first, second, third, and fourth central moments?
Senior
What is the law of the unconscious statistician and how do you use it to compute the expectation of a function of a random variable?
Senior
What is conditional expectation, and can you explain the law of total expectation?
Senior
Can you explain the law of total variance?
Senior

Bayes Theorem & Inference

Explain Bayes' Theorem and its applications, for example in machine learning or spam detection.
Mid
How do you update probabilities using the concepts of prior, likelihood, and posterior?
Mid
Can you explain the formula of Bayes' Theorem and the meaning of each variable (prior, likelihood, evidence/marginal, posterior)?
Mid
Which component of Bayes' theorem is proportional to likelihood × prior?
Mid
What is the base-rate fallacy and why does ignoring the prior lead to wrong conclusions?
Mid
How do you derive Bayes' theorem from the definition of conditional probability?
Mid
Can you explain the Bayesian approach to probability, contrasting it with the frequentist view?
Senior
Can you express Bayes' theorem in odds form using the likelihood ratio?
Senior
How does sequential Bayesian updating work as multiple pieces of evidence arrive?
Senior

Counting & Combinatorics

What is the difference between a permutation and a combination, and when would you use each?
Junior
Can you explain the multiplication (counting) principle in combinatorics?
Junior
How would you approach challenging combinatorial probability questions involving coin tosses, dice throws, cards, and balls?
Mid
What is the difference between sampling with replacement and without replacement when counting outcomes?
Mid
Can you explain the birthday problem and why the probability of a shared birthday is higher than intuition suggests?
Mid

Diagnostic Testing & Applied Bayes

Given the prevalence of a disease, and the sensitivity and specificity of a test, if a person tests negative, what is the probability they actually have the disease?
Mid
Explain how to calculate conditional probabilities in real-world scenarios, such as in disease testing.
Mid
Given an outcome, how would you find the probability of a coin being biased?
Mid
How would you determine the probability that a randomly selected defective item was produced by a specific machine, using Bayes' Theorem?
Mid
A model detects spam with 99% true positive rate but a 2% false positive rate, and only 10% of emails are spam — what is the probability that an email flagged as spam is actually spam?
Mid
What do sensitivity, specificity, and positive predictive value mean in a diagnostic-test problem?
Mid
Can you explain the 'False Positive Paradox' or 'Prosecutor's Fallacy' in the context of a medical test?
Senior
How might you use Bayesian methods to conceptually improve the performance of a spam classifier?
Senior

Simulation & Probability Puzzles

Can you explain the Monty Hall problem and its solution, focusing on the probabilistic reasoning?
Mid
How would you set up a Monte Carlo simulation to estimate the value of pi?
Mid
How are Monte Carlo simulations used to approximate probabilities and expectations?
Senior
How can you generate fair odds using a coin with an unknown bias toward heads or tails?
Senior
Why do Monte Carlo estimates become more accurate as the number of simulations increases, and how fast does the error shrink?
Senior
Can you explain the coupon collector problem and how you'd compute the expected number of purchases to collect all items?
Senior
Can you explain the gambler's ruin problem and the reasoning behind it?
Senior

Probability & Bayes' Theorem

Random Variables & Distributions

How do you define and use random variables in probability?

What is the difference between discrete and continuous probability distributions?

What is a probability density function (PDF)?

What is the role of the cumulative distribution function (CDF)?

How would you explain a probability distribution to a layperson?

What is the difference between a PDF and a CDF, and how are they related?

How do Probability Mass Functions (PMF) and Probability Density Functions (PDF) differ?

What are probability distributions and their importance in machine learning?

How could a probability distribution be non-normal, and can you provide an example scenario such as a uniform distribution?

How do you find the distribution of a function of a random variable?

Probability Fundamentals

What are independent and dependent events in probability?

What is probability, and how is it used in machine learning?

Can you define the terms 'sample space' and 'event' in probability?

What are the basic rules of probability, including the addition rule, multiplication rule, and complement rule?

What are the Kolmogorov axioms of probability?

What is the difference between mutually exclusive events and independent events?

What is the Law of Large Numbers?

Can you explain the inclusion-exclusion principle for the probability of a union of events?

What is the difference between the Law of Large Numbers and the Central Limit Theorem?

Conditional Probability & Independence

How would you calculate the probability of 'at least one' event occurring using the complement rule?

Can you explain the differences between joint, marginal, and conditional probabilities?

Explain why P(A|B) is not necessarily equal to P(B|A).

What is the law of total probability and how do you use it to partition a sample space?

Can you explain the chain rule (general multiplication rule) for probability?

How can assuming independence in probabilistic models lead to inaccuracies?

What is the difference between independence and conditional independence?

Common Distributions

What are the characteristics of a normal distribution?

What are z-scores and standardization, and what does the 68-95-99.7 rule tell you about a normal distribution?

When would you model a scenario with a uniform distribution, and what are its mean and variance?

When would you use a Binomial distribution, and what are its key properties such as mean and variance?

Explain the utility of the Binomial distribution in machine learning.

How does the Poisson distribution differ from the Binomial distribution?

What is the relevance of the Bernoulli distribution in machine learning?

When do you use the binomial, Poisson, and normal distributions?

How would you compute the expected number of trials until the first success?

When does the geometric distribution arise, and what are its mean and variance?

Can you explain the Central Limit Theorem and its significance in machine learning?

What is the memorylessness property of the exponential distribution and what does it mean?

When can the Poisson distribution be used to approximate the Binomial distribution?

Expectation Variance & Moments

What is expected value and how is it used?

Is expected value a value you actually expect to see?

What is the difference between probability and likelihood?

Can you define expectation, variance, and covariance?

Can you explain the linearity of expectation?

What are the key properties of variance, for example how variance behaves for sums of random variables?

What is the difference between covariance and correlation?

How would you decide whether a game or bet is worth playing based on its expected value?

What is a central moment of a probability distribution of a random variable about its mean, and can you describe the zeroth, first, second, third, and fourth central moments?

What is the law of the unconscious statistician and how do you use it to compute the expectation of a function of a random variable?

What is conditional expectation, and can you explain the law of total expectation?

Can you explain the law of total variance?

Bayes Theorem & Inference

Explain Bayes' Theorem and its applications, for example in machine learning or spam detection.

How do you update probabilities using the concepts of prior, likelihood, and posterior?

Can you explain the formula of Bayes' Theorem and the meaning of each variable (prior, likelihood, evidence/marginal, posterior)?

Which component of Bayes' theorem is proportional to likelihood × prior?

What is the base-rate fallacy and why does ignoring the prior lead to wrong conclusions?

How do you derive Bayes' theorem from the definition of conditional probability?

Can you explain the Bayesian approach to probability, contrasting it with the frequentist view?

Can you express Bayes' theorem in odds form using the likelihood ratio?

How does sequential Bayesian updating work as multiple pieces of evidence arrive?

Counting & Combinatorics

What is the difference between a permutation and a combination, and when would you use each?

Can you explain the multiplication (counting) principle in combinatorics?

How would you approach challenging combinatorial probability questions involving coin tosses, dice throws, cards, and balls?

What is the difference between sampling with replacement and without replacement when counting outcomes?

Can you explain the birthday problem and why the probability of a shared birthday is higher than intuition suggests?

Diagnostic Testing & Applied Bayes

Given the prevalence of a disease, and the sensitivity and specificity of a test, if a person tests negative, what is the probability they actually have the disease?

Explain how to calculate conditional probabilities in real-world scenarios, such as in disease testing.

Given an outcome, how would you find the probability of a coin being biased?

How would you determine the probability that a randomly selected defective item was produced by a specific machine, using Bayes' Theorem?

A model detects spam with 99% true positive rate but a 2% false positive rate, and only 10% of emails are spam — what is the probability that an email flagged as spam is actually spam?

What do sensitivity, specificity, and positive predictive value mean in a diagnostic-test problem?