# 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.

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# 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?

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

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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?

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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?

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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?

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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.)

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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?

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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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