During the course of a day a machine turns out two items, one in the morning and one in the afternoon. The quality of each item is measured as good (G), mediocre (M), or bad (B). The longrun fraction of good items the machine produces is 1 2, the fraction of mediocre items is 1 3, and the fraction of bad items is 1 6.
(a) In a column, write the sample space for the experiment that consists of observing the day’s production.
(b) Assume a good item returns a profit of $2, a mediocre item a profit of $1, and a bad item yields nothing. Let X be the random variable describing the total profit for the day. In a column adjacent to the column in part (a), write the value of this random variable corresponding to each point in the sample space.
(c) Assuming that the qualities of the morning and afternoon items are independent, in a third column associate with every point in the sample space a probability for that point.
(d) Write the set of all possible outcomes for the random variable X. Give the probability distribution function for the random variable.
(e) What is the expected value of the day’s profit?

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