<p>1. SouthPart Autos offers a variety of luxury options on its cars. Because of the 6 to 8 week waiting period for the customer orders, Ben Stolen, the dealer stocks his cars with a variety of options. Currently Mr stolen who prides himself on being able to meet his customers needs immediately, is worried because of an industry wise shortage of cars with V-8 engines. Stolen offers the following luxury combinations</p><p>1. V-8 engine, electric Sun roof, halogen headlights</p><p>2. Leather interiors, , power door locks, stereo cassette deck</p><p>3. Halogen headlights, V-8 Engine, Leather interior</p><p>4. Stereo cassette deck, V8 engine , power door locks</p><p>Stolen thinks that combination2,3 &4 have an equal chance of being ordered but the combination 1 is lower as likely to be ordered as any of these. What is the probability that any customer ordering a luxury car will order with V-8 engine?</p><p>2. A recent study of how Americans spend their leisure time surveyed workers employed more than 5 years. They determined the probability an employee has 2 weeks of vacation time to be .45, one week of vacation time to be .10 and 3 or more weeks to be .20. Suppose 20 workers are selected at random </p><p> a) What is the probablilty that only one worker has one week of vacation time?</p><p> b) What is the probability that atmost 2 workers have 3 or more weeks of vacation time?</p><p>3. Given the binomial distribution with n = 2.5 and p = .032, use the poisson approximation to the binomial to find. A) P[r=3] B)P[r<=1]</p><p>4. Median is a value x such that P[X<x]<=1/2 and P[X<=x] >=1/2. Mode of a distribution of X is a value x that maximizes f(x). If there is only one such x, it Is called the mode of the distribution</p><p>Find the median and of the pdf/p.m.f of X where </p><p>I. F(x)= (1/2)^x , x= 1,2,3,4…..</p><p>= 0 otherwise</p><p>II. f(x)= (½) (x^2 )(e^-x), 0<x<∞</p><p>= 0 elsewhere</p><p>III. f(x) = 1/(b-a) if a<x<b</p><p>= 0 otherwise</p><p>5. Ashok tells the truth with probability 0.5 and kalpana with probability 0.4</p><p>I. If they make the same statement, what is the chance that the statement is true? II. If Ashok makes a statement and kalpana says that ashok lies, what is the probability that ashok told the truth?</p><p>(Hint: Use Bayes theorem)</p><p>6. Certain missile components are shipped in lots of 12. Three components are selected from each lot and a particular lot is accepted if one of the three components selected is defective. What is the probability that a lot will be accepted if it contains 5 defectives (Hint use hypergeometric distribution)</p>