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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?

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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 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?

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

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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 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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