How to Answer Probability Questions
In this lesson, we'll teach you how to answer the most common types of statistics questions.
Conceptual questions
Conceptual statistics questions test your ability to contextualize statistical concepts outside of theory. The main tip is to clearly communicate and use real-world examples to supplement the theory and clarify the conceptual explanation.
Say you’re asked, “What is Bayes theorem?” A strong interview answer includes a concise definition, walkthrough of its formula, and a real-world example of its application. You can explain how Bayes theorem can be used to understand the probability of false positives or negatives in medical tests and provide an example using realistic numbers.
Numerical and applied questions
Numerical and applied statistics interview questions allow interviewers to evaluate your approach and execution of statistical techniques and formulas. In addition to mathematical accuracy, interviewers are assessing your ability to communicate your approach in an organized way.
Below, we describe a 6-step framework for answering numerical and applied questions.
- Step 1. Define the problem. Ask clarifying questions and present the problem statement.
- Step 2. Identify assumptions and variables. Identify relevant outcomes and any conditions or criteria.
- Step 3. Apply the statistical technique or formula. Use standard statistics/math terminology where possible.
- Step 4. Check your work. Check your reasoning and calculations to ensure accuracy.
- Step 5. Re-visit the problem scope. Interpret the result in the context of the problem.
- Step 6. Check in with the interviewer. Be open to feedback and constructive criticism.
As you practice with this framework, remember to review Rubric for Statistics & Experimentation Questions to understand how interviewers are evaluating your answer.
Say you’re given this scenario: “Suppose a factory produces light bulbs, and two machines, A and B, are responsible for producing them. Machine A produces 60% of the bulbs while machine B produces the remaining 40%. Machine A produces defective bulbs at a rate of 5% while machine B produces defective bulbs at a rate of 3%. If a randomly selected bulb is defective, what is the probability that it was produced by machine A?”
An effective way to answer this numerical question is to follow the framework outlined below.
Step 1: Define the problem
When appropriate, ask clarifying questions and define the scope before starting to think about the solution. Clarifying questions are particularly relevant when presented with ambiguous case study questions. Pay attention to any conditions, constraints, or assumptions mentioned in the question. Missing important details can lead to incorrect solutions.
Then, present the problem statement.
In the lightbulb example, the question is relatively straightforward, so clarifying questions are less relevant. The interviewer is mainly testing whether you are able to identify and implement the correct statistical technique. Therefore, you could say,
“The problem statement here is to calculate the probability that a bulb is produced by Machine A, given that it is randomly selected and defective, i.e. conditional probability. Therefore, we should use the Bayes theorem statistical technique.”
Step 2: Identify assumptions and variables
Clearly identify the outcomes of interest and any conditions or criteria specified in the question. Identify any data preprocessing or validation steps, if relevant.
In the lightbulb example, you could say,
“The main assumption is conditional independence. Bayes theorem assumes that the events or variables involved are conditionally independent. This means that the probability of one event occurring is not affected by the occurrence of another event, given the value of a third event.
Our variables are:
P(A) = Probability that Machine A produces the bulb = 60% = 0.6
P(B) = Probability that Machine B produces the bulb = 40% = 0.4
P(D) = Probability that a bulb is defective
P(D|A) = Probability that a bulb is defective given it was produced by A = 5% = 0.05
P(D|B) = Probability that a bulb is defective given it was produced by B = 3% = 0.03
P(A|D) = this is what we need to calculate
P(D) = P(D|A)P(A) + P(D|B)*P(B)
= 0.050.6 + 0.030.4
= 0.042”
Step 3: Apply the statistical technique or formula
Use the standard statistics or math terminology where possible. If you don’t remember the name of a theorem or technique, briefly describe it to ensure the interviewer understands.
If there are multiple valid approaches, state them briefly and incorporate feedback from your interviewer to select the preferred approach. Discuss the pros and cons of each method. For example, if you’re given a probability question, you can discuss the different solutions, such as a simulation vs. analytical solution. Simulations are usually more computationally intensive but require fewer assumptions to be valid. Then, you could compare the two by reflecting that analytical solutions are typically faster to compute and more precise, but they require more assumptions to be valid.
Consider the feasibility, practicality, and potential risk of solutions and how they would be implemented in a real-world scenario. If helpful, utilize a whiteboard to visualize your thought process.
In the lightbulb example, you could say,
“We’ll implement the Bayes theorem calculation:
P(A|B)P(B) = P(B|A)P(A)
Now, we’ll plug in the values:
P(A|D) = P(D|A)*P(A)/P(D)
= 0.05*0.6/0.042
= 5/7”
Discussing multiple approaches isn’t relevant here, since there usually aren't better alternatives to using Bayes Theorem for conditional probability questions.
Step 4: Check your work
After obtaining your solution, double-check your calculations and reasoning to ensure accuracy. Avoid the pitfall of skipping steps in calculations, not showing your thought process and just stating the final answer.
In the lightbulb example, you could say,
“Now, let’s double-check our calculations. One way to check the math is to calculate:
P(B|D) = P(D|B)*P(B)/P(D)
= 0.03*0.4/0.042
= 2/7
P(A|D) + P(B|D) = 5/7 + 2/7 = 1. This is correct because the bulb can only be produced by Machine A or Machine B. These probabilities should add to 1. This calculation validates our result above.”
Step 5: Re-visit the problem scope
Once you’re happy with the solution, interpret the result in the context of the problem. Ensure that the results answer the question posed in the problem.
In the lightbulb example, you could say,
“As defined previously, we need to calculate the probability that a bulb is produced by machine A, given that it is randomly selected and defective.
Given a randomly selected bulb is defective, the probability that it is produced by Machine A is 5/7. This makes sense, since Machine A produces defective bulbs at a higher rate of 5%, while Machine B produces defective bulbs at a rate of 3%.”
Step 6: Check in with the interviewer
Be prepared to discuss your solution further if the interviewer has follow-up questions or wants to explore alternative approaches. Be open to feedback and constructive criticism.