You may use calculators and tables on this exam.
1. As the cars roll in on I-Day, we note that 35% are minivans. Also, 25% of the vehicles already have Naval Academy stickers on them.
If 10% of the vehicles are minivans with stickers, what fraction are non-minivans without stickers?
2. Suppose a plebe class contains 900 men and 150 women. We note that 20% of the male plebes arrive with no family, while 8% of the women arrive with no family.
(a) Are "arriving with no family" and "male" independent? Why?
(b) Are "arriving with no family" and "male" mutually exclusive? Why?
(c) Given that a plebe arrived with no family, what is the probability the plebe is a male?
3. One plebe has wandered off. There is a 25% chance he is in the head, a 35% chance he is in the wrong line
and a 40% chance that he walked off with the wrong company. We go looking for him. If he is in the head,
there is a 50% chance we'll find him. If he is in the wrong line, there is a 30% of finding him.
If we went with the wrong company, there is a 20% of finding him.
(a) If we search all 3 locations, what is the chance of finding him?
(b) If we have searched all 3 locations without finding him, what is the chance that he went off with the wrong company?
(c) If, instead of searching all 3 locations, we were to pick just one location to search, which one should it be? Why?
4. (a) Only about 20% of the plebes require size Small. We have 3 Small shirts left and 10 more plebes to serve.
What is the probability that we will have enough Small shirts?
(b) Suppose we have run out of Small shirts. On average, how many plebes can we serve before someone needs a Small?
(c) Suppose we have served 10 plebes and no one has asked for a Small yet. What is the probability that we can serve 5 more without being asked for a Small?
5. One of our brighter employees went to the back and got 10 Large and 6 Small shirts. However, he mixed them up in the box.
If we pull 5 shirts from the box, what is the chance that we will get (exactly) 3 Smalls?
6. We need to make a trip to get a special size about every 20 minutes.
(a) What is the chance we make (exactly) 3 trips in one hour?
(b) What is the chance of making no trips in an hour?
7. Suppose that pants lengths (inseams) have a normal distribution with mean 32 and variance 4.
(a) What fraction of mids have inseams between 31 and 32?
(b) What size pants (inseam) will be large enough for 99% of mids?
(c) The shortest 25% of mids have inseams up to what size?
8. Suppose that the time spent waiting in a given line has an exponential distribution with mean 15 minutes.
(a) What is the probability of finishing a line within 5 minutes?
(b) 90% of plebes wait less than what time? (Answer will be larger than 15 minutes.)
(c) If the plebes have to wait in 3 lines in succession, their total waiting time will be the sum of the 3 (independent) waiting
times. What is the chance that a plebe can finish all 3 lines within 30 minutes?
9. Suppose that the weight of each plebe's gear has mean 30 lbs and variance 25.
(a) If we pile 20 plebes' gear on a cart, what is the mean weight of the cart?
(b) What does the load capacity of the cart have to be so that we are 99% sure it is not overloaded? (Assume that the Central Limit Theorem holds.)
10. Suppose that head circumferences have a uniform distribution from 18 to 24 inches. For simplicity, suppose that hats are circles with these circumferences.
(a) What is the median area of a hat?
(b) What is the mean area of a hat?
11. Suppose a continuous random variable, X, is defined on 0<x<5 and has cdf F(x) = x3/125
(a) What is the probability that X lies between 1 and 3?
(b) What is the median of X?
(c) What is the mean of X?
12. Suppose that X and Y are independent. Let X have a binomial distribution with N=6 and p=0.2. Let Y have a Poisson distribution with mean 1.5.
If W=X+Y, what is the probability that W=1?