1. (a) At a fancy restaurant, you have your choice of 3 salads, 5 entrees and 2 desserts. How many different meals are possible?
(b) A section contains 12 men and 3 women. If 4 mids are chosen at random to go to the board, what is the probability that 2 of the 4 will be women?
2. (a) When I go on travel, there is a 39% chance I will get a foreign rental car. If I get a foreign car, there is a 70% chance I can't figure out the headlights.
If I get a domestic rental car, there is only a 22% chance I can't figure out the headlights. What is the chance I will have trouble with the headlights?
(b) In (a), if I don't have trouble with the headlights, what is the chance that I got a foreign rental car?
(c) If A is the event I have trouble with the headlights and B is the event I get a foreign rental car, are A and B independent? Why?
3. My daughter reports that she cannot find her sweater. The following give the probability of it being in each of 3 places and the probabilty of finding it there, if we search.
Loc A1 A2 A3 Prob(Ai) 0.5 0.3 0.2 Prob(D|Ai) 0.1 0.3 0.3
(a) If we are to search only 1 place, where should we look?
(b) If we do not find the sweater, what is the probability that it is in location A1?
4. Suppose the probability density function, f(x), for a random variable is given by f(x) = 2-2x, 0<x<1.
(a) Find E(X)
(b) Find Var(X)
5. Suppose the cdf for a random variable is given by F(x) = x^3, 0<x<1
(a) Find Prob(.1 < X < .2)
(b) Find Prob(X > 0.5)
6. (a) Grand prize winners occur about twice an hour. What is the probability of seeing at least one winner in a 10 minute period?
(b) The game is played by drawing 4 beads from a jar containing 6 black and 3 red. You win if you get at least 3 black beads. What is the probability of winning?
(c) My friend only wants to win his game once. If he has enough money for 6 plays and the probability of winning on
any one play is 0.32, what is the probability he will win (at least) once?
7. Suppose X has a geometric distribution with p=0.4 and Y has geometric distribution with p=0.5. Let X and Y be ind
and W=X+Y. Find Prob(W=3)
8. Suppose that the weight of cereal boxes has a normal distribution with mean 15 oz and standard deviation 1.3 oz.
(a) Prob(Wt < 14.5)
(b) Prob(14 < Wt < 16)
(c) If Prob(Wt < c) = 0.75, find c.
9. A 40 question multiple choice exam has 4 choices for each question. What is the highest score we can get with probability=0.9, if we guess?
10. Scrap lengths of wood have an exponential distribution with mean 3 feet.
(a) What fraction of the pieces will be at least 2 feet long?
(b) I want a box that is long enough to hold 80% of the pieces. How long should it be?
(c) If I want to place 4 pieces end to end, what is the probability that the total length will be as much as 16 feet?
11. Let X be the amount of plastic extruded from a machine. Suppose X has a uniform distribution on (3,4). Let S be the side of a square whose area is X.
(a) Find Prob(S > 1.9)
(b) Find E(S)