Math for Quantitative Finance

Interview

Welcome to your probability interview! In a quantitative finance interview, one of the most important things is to explain your thoughts. A good thought process is worth more than a correct answer with no ability to articulate your reasoning. This quiz is no different, as we’ll be breaking down two problems into their logical steps.

Interview

For the first problem, consider the following:

A \(3\times 3\times 3\) cube is painted red, and then cut into 27 \(1\times 1\times 1\) cubes. One of these cubes is chosen at random, and rolled like a die. What is the probability that it lands with a painted side facing up?

Interview

First of all, what is the total area that has been painted red?

                 

Interview

Great, 54 faces are painted red. Next, we could figure out how many \(1\times 1 \times 1\) cubes have 1 face painted, 2 faces painted, etc. Is this step necessary?

                 

Interview

To avoid all that computation, we can simply look at how many faces are red out of all of the faces of the \(1\times 1 \times 1\) cubes. What does this give for the final probability?

                 

Interview

For the second problem, consider the following:

There are two boxes, each with three balls. Box A has two red balls and one blue ball; Box B has one red ball and two blue balls. I randomly choose a box, and randomly draw a red ball. If I then draw another ball from the same box, what is the probability that it is also red?

Interview

First of all, what is the a priori probability of choosing box A and drawing a red ball? ("A priori" here meaning: only given the conditions that Box A has two red balls and one blue ball and Box B has one red ball and two blue balls, before any drawing has taken place.)

                 

Interview

Given the previous answer (and the fact you start by drawing a red ball), what is the conditional probability that your initial draw was out of box A?

                 

Interview

Wrapping things up, what is the answer to the probability question below? Keep in mind that there is no replacement.

There are two boxes, each with three balls. Box A has two red balls and one blue ball; Box B has one red ball and two blue balls. I randomly choose a box, and randomly draw a red ball. If I then draw another ball from the same box, what is the probability that it is also red?

                 
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