1.(a) The nominating committee has located 6 men and 4 women who are willing to serve. If we need 4 members of the board of trustees, how many boards can be formed?
(b) What is the probability that the board of trustees will contain exactly 2 women?
(c) How many sets of President, Vice-President, Treasurer are possible?
2. (a) 82% of all butterflies are spotted. 37% of butterflies prefer yellow flowers. If 19% of butterflies are spotted and prefer yellow flowers, how many butterflies are neither spotted nor prefer yellow flowers?
(b) A plant has two assembly lines. There is a 57% chance that the inspector will draw a sample from line A and a 43% chance he will draw it from line B. 17% of the items on line A are defective and 29% of the items on line B are defective. What is the probability that the inspector will draw a defective item?
(c) On a plane trip, suppose that the probability of a lousy movie is 44% and the probability of a bad meal is 71%. If these are independent, what is the probability of both a lousy movie AND a bad meal?
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 search all 3 locations, what is the probability of finding the sweater?
(b) If we do not find the sweater, what is the probability that it is in location A1?
4. Suppose the probability mass function for a random variable is given below.
(a) Find E(X)
(b) Find Var(X)
x p(x) 0 0.3 1 0.3 2 0.2 3 0.2
5. Suppose the cdf for a random variable is given by F(x) = x^2, 0<x<1
(a) Find Prob(.1 < X < .2)
(b) Find Prob(X > 0.5)
6. The probability of winning a certain carnival game is 0.32.
(a) Find the probability of winning 3 times in 5 plays.
(b) We have to win 3 times to get the bear. On average, how long will this take?
(c) What is the probability that we will win the bear within 5 plays?
7. Suppose X has a binomial distribution with N=4, p=0.45 and Y has a geometric distribution with p=0.63. X and Y are ind. Let W=X+Y. Find Prob(W=2)
8. Suppose that SAT scores are normally distributed with mean 1050 and standard deviation 110.
(a) Find Prob(SAT>1200)
(b) Prob(1000 < SAT < 1300)
(c) If Prob(SAT > c) = 0.3, what is c?
9. A 40 question multiple choice exam has 4 choices for each question. Find the probability of more than 15 right if we simply guess.
10. The time between arrivals at an Air Traffic Control Center has an exponential distribution with mean 2.5 minutes.
(a) Find the probability that the time to the next arrival is less than 1.5 minutes.
(b) What is the longest time we can have with probability 0.8?
(c) If I'm away from my station for 5 minutes, what is the probability that 3 or more planes arrive while I'm gone?
11. Suppose we make circles whose radii have a uniform distribution on (1,2). Let A be the area of the circle.
(a) What is the Prob(A < 2 pi)?
(b) What is E(A)?