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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.

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?

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

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?

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?

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 draw a red ball. If I then draw another ball from the same box, what is the probability that it is also red?

First of all, what is the a priori probability of choosing box A and drawing a red ball?

Given the previous answer, what is the conditional probability that you chose box A originally?

Wrapping things up, what is the probability that the second ball you draw from the same box is also red? Keep in mind that there is no replacement.


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