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# Probability Mass Function and Room Occupany

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An entrepreneur owns a small bed and breakfast inn with five rooms. Let the random variable X be the number of rooms that are occupied by guests each night. A portion of the probability mass function appears below.

X 0 1 2 3 4 5
P(x) 0.2 0.25 0.3 0.15

a. Define the given probability mass function completely by filling in the missing probabilities, knowing that the probability of all five rooms being occupied by guests on any given night (i.e. having no vacancies) is three times larger than the probability of having no rooms occupied by guests.

b. Based on the above probability distribution, what is the expected number of rooms that will be occupied by guests each night?

c. What is the probability that on a randomly selected night, the inn will have at least three rooms occupied by guests?