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(3 points) Suppose an employer randomly selects 5 new employees from a total of 8 applicants, 1 men and 7 women. Let X be the number of women who are hired. Find the following. \"X = 0x = P(X=4)= (2 points) A quality-control engineer inspects a random sample of 6 batteries from each lot of 24 car batteries ready to be shipped. If such a lot contains 5 batteries with slight defects, what are the probabilities that the inspector's sample will contain: a) 1 of the batteries with defects? b) 4 of the batteries with defects? (4 points) Given that X is a hypergeometric random variable, compute P(X = x) for each of the following cases: a) N = 11, n = 3, M = 3, x = 3 P(X = 3) = b) N = 8, n = 5, M =3, x =0 P(X = 0) = c) N = 4, n = 2, M = 2, x = 0 P(X = 0) = d) N = 5, n = 2, M = 2, x =1 P(X = 1) =(2 points) Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 10 per hour. a) Determine the probability that over the next hour, only one car will arrive. P(X = 1) = b) Compute the probability that in the next 7 hours, more than 25 cars will arrive. P(X > 25) = (1 point) The average number of patients admitted per day to the emergency room of a small hospital is 2.5. If, on any given day, there is/are only 1 bed(s) available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? P(The hospital won't have enough beds) = (1 point) A statistics professor finds that when she schedules an office hour for student help, an average of 1.6 students arrive. Let X be the number of student arrivals in an office hour. Find the probability that in a randomly selected office hour, the number of student arrivals is 5. P(X=5)= (4 points) The following density function describes a continuous random variable X. 24}: if 12 10) = (3 points) The following density function describes a continuous random variable X. f(x) = 1 - N X if 0 1) = b) P(X