Finite Math Final Exam Dec. 14, 2000 F Track R. Hammack Name: ________________________ Score: _________

(1) Solve the following systems of equations:
(a)
 x - 2y = -8 3x - 6y = -24 x - y = -5

(b)

 x + y + z = 2 x + y - z = 0 x + y + 2z = 3

(2) Maximize of the objective function P = x + 5y subject to the following inequalities:
 x + y ≤ 6 x - y ≤ 4 x + y ≥ 2 x ≥ 0

Notice that this linear programming problem is not in standard form. Thus the simplex method will not work. Solve it by graphing.

(3) Use the simplex method to maximize P = x1 + 3x2 + 4x3 + x4, subject to the following constraints:

 2x1 + x2 + 2x3 + 4x4 ≤ 2 2x1 + 2x2 + 2x3 + 2x4 ≤ 3 x1, x2, x3, x4 ≥ 0

(4) A department consists of 7 men and 5 women.

(a) A committee of 4 people is chosen from this department. In how many ways can this be done?

(b) A chair and a treasurer are chosen from this department. In how many ways can this be done if it is required that the two positions not be held by two people of the same gender?

(5) Two cards are dealt off a well-shuffled deck.

(a) You win \$1 if the two cards are of different suits. Find the probability of your winning.

(b) You win \$1 if either both cards are red or both cards are black. Find the probability of your winning.

(6) A fair coin is tossed 6 times. Find the probabilities of the following events:

(a) The first 3 tosses are all heads.

(b) It is not the case that the first 3 tosses are all heads.

(c) There are the same number of heads as tails.

(d) There are more heads than tails.

(8) A box contains 5 lettered blocks, as illustrated.
You make two random draws from the box, with replacement. (I.e. you draw a block, put it back in the box, then draw a block again.)

 M

 A
 C
 O
 N

(a) What is the probability that both draws were vowels?

(b) What is the probability that the second draw is an M?

(c) What is the probability that either the first draw was a vowel or the second draw was an M?

(d) What is the probability that both draws were vowels and the second draw was an A?

(9) Find the optimal strategies for each player in the following matrix games. Determine if each game is fair, and if not, which player has the advantage.
(a)

 [ -3 -4 2 1 ] 6 5 -2 -3

(b)

 [ -4 -8 ] -1 3

(10) On a given day, a student is either healthy or sick. Of the students who are healthy today, 90% will be healthy tomorrow. Of the students who are sick today, 50% will be sick tomorrow. Today 80% of the students are healthy.

(a) What percentage of the students will be healthy tomorrow? What percentage will be sick?

(b) Assuming this trend continues, what will the percentage of healthy students be in the long run?