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# Interview Questions for the Quantitative Trader Job

## A series of interview questions that may appear in quantitative trader job interviews.

O
OpenQuant
2023-02-23

Quantitative trading interviews typically consist of a series of brainteasers/puzzles, mental math questions, and statistics problems. These questions can vary from firm to firm but are generally asked during multiple rounds of interviews, beginning with an online assessment. These questions tend to be particularly frequent in entry-level and internship roles.

For mental math questions, the best way to prepare is to practice repeatedly solving these questions under a designated time constraint. Typically these questions are administered in the form of an online assessment, where you'll have a few minutes to solve as many of these questions as you can. Optiver, for example, gives you 8 minutes to solve 80 mental math questions that can include integers, decimals, and fractions. A good resource to practice quantitative trading mental math questions is the Optiver Game on OpenQuant. We recommend a score of at least 55 points if you would like to be considered a competitive applicant.

For the statistics portion of the interview, it's best to revisit some of your textbooks from any statistics courses you took in college. One great resource is First Course in Probability by Sheldon Ross. This book gives great explanations of the mathematics of probability theory and also has numerous practice problems to choose from.

Outside of the resources mentioned above, the best way to get good at brainteasers and statistics problems is by solving more variations of these types of problems. In this article, we'll cover a handful of practice problems that you could see appear in your next quantitative trading interview. Scroll to the bottom to find detailed answers with explanations.

### Questions

1. SAT Math (Statistics)
2. Test Scores (Mathematics)
3. Coin Toss (Statistics)
4. Complete the Series (Mathematics)
5. Weather (Statistics)

## Question #1 - SAT Math

Assume you are taking the mathematics portion of the SAT, in which scores range from 200 to 800. For this portion of the exam, scores are normally-distributed with a mean of around 500. When the test score results are released, your friend tells you that you scored more than 1 standard deviation away from the mean. Given this information, what is the probability you scored more than 2 standard deviations from the mean?

a) 16%

b) 9%

c) 50%

d) 27%

## Question #2 - Test Scores

The average of 6 tests scores is 81. After adding a seventh test score, the mean is now 83. What was the seventh score?

a) 88

b) 97

c) 92

d) 95

## Question #3 - Coin Toss

A fair coin is tossed 6 times. What is the probability of getting exactly 4 heads?

a) 5/32

b) 15/64

c) 13/64

d) 5/16

## Question #4 - Complete the Series

What number comes next in each series?

Part A: 2, 4, 7, 11, 16, _

a) 21

b) 22

c) 23

d) 24

Part B: 81, -54, 36, -24, _

a) -12

b) 12

c) 16

d) 18

## Question #5 - Weather

You are getting ready to plan a four-day vacation trip, but are concerned about the weather. At the resort, 80% of days are sunny, and each day is independent from the other days. In order for you to have a good time, at least 3 of the 4 days must be sunny. What is the probability that you have a good time?

a) 64/125

b) 128/625

c) 256/625

d) 512/625

### Question #1

The correct answer to this question is choice a (16%).

According to the normal model, 68% of the data falls within 1 standard deviation from the mean. This means that the probability that a value is outside 1 standard deviation is 0.32 (1 - 0.68). Furthermore, 95% of the data falls within 2 standard deviations from the mean, and therefore the probability of a value being outside two standard deviations is 0.05. 0.05 / 0.32 ~ 0.16.

### Question #2

The correct answer to this question is choice d (95).

In this problem we are told that the average of 6 test scores is 81. Mathematically, we can express this as X / 6 = 81. Here, X represents the sum of the six previous test scores. Solving for X gives us the value 486. Next, we want to find the value of the seventh test score. Once again, we can express this as (486 + Y) / 7 = 83. Here, Y represents the seventh test score. Solving for Y gives us 95.

### Question #3

The correct answer to this question is choice b (15/64).

First, we can calculate the total number of possibilties from tossing the coin six times. Since there are two outcomes for each coin flip, the number of total possibilities is 2^6 = 64. To calculate the number of ways in which we can get four heads, we can use the combination formula n!/(r!(n-r)!). The total number of objects is 6 and the number of objects we're choosing to be heads is 4, so we get 6!/(4!(6-4)!) = 6!(4!*2!) = 15. Therefore, the answer is 15/64.

### Question #4

The correct answer to part A is answer choice b (22) and the correct answer to part B is answer choice c (16).

For part A we notice that the rate of increase for each consecutive number is increasing (+2 from 2 to 4, +3 from 4 to 7, +4 from 7 to 11). Mathematically, we can represent this sequence as n(n+1)/2 + 1.

For part B, let's imagine that we reversed the sequence. Doing so, we notice that each consecutive number is the (current number + 1/2 * current number) * -1. For example, (-24 + 1/2 * -24) * -1 = 36. Therefore, the answer to this question will be a number, that when added with half of itself and multiplied by -1, gives us the number -24. Therefore the answer is 16.

### Question #5

Since the question wants us to find the probability that at least 3 of the 4 days are sunny, we can find the probability that exactly three days are sunny + probability that exactly 4 days are sunny. The probability that three days are sunny is the number of ways that you can choose those 3 days out of 4, multiplied by 0.8 for the sunny day and multiplied by 0.2 for the rainy day. Mathematically, this becomes:

(4C3) * (.8^3) * (0.2^1) = 0.4096 where (4C3) is the combination formula with n=4 and r=3.

Similarly, we can calculate the probability that it rains all four days as:

(4C4) * (0.8^4) * (0.2^0) = 0.4096 