Math 070, Practice Exam 2 April 25, 205 The Honor Code is in effect for this examination. All work is to be your own. You may use your Calculator. The exam lasts for 50 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e)... 3. (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)... 5. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) (e)... 7. (a) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice 8. 9. 0.. Total
Multiple Choice.(6 pts.) The probability distribution of a random variable X given below, find E(X) and the standard deviation σ(x). k Pr(X = k) - 2 0 8 6 2 6 3 4 (a) 4 (b) 6 (c) 7 6 (d) 4 (e) 3 8 2.(6 pts.) A basketball player with career field goal percentage of 80% takes 00 shots in a row. If his probability of making a basket on every shot is 0.8, what is the longest run of misses that you would expect in the data? (a) 3 (b) 3 (c) 0 (d) 6 (e) 20 2
3.(6 pts.) Given a sequence with two values, success (S) and failure (F), with N s success and N f failures, let X denote the number of runs (of both S s and F s). Wald and Wolfowitz determined that for a random sequence of length N with N s success and N f failures (note that N = N s + N f ), the number of runs has mean and standard deviation given by E(X) = µ = 2N sn f (µ )(µ 2) +, σ(x) =. N N The distribution of X is approximately normal if N s and N f are both bigger than 0. What is the value of X for the following sequence of success and failures and what is its Z-score (the number of standard deviations from the mean)? S F S S S F F S S S S S F F F F S S F S (a) X = 9, Z 0.77 (b) X = 5, Z 2.69 (c) X = 5, Z 2.69 (d) X = 9, Z 0.77 (e) X = 0, Z 0.29 3
4.(6 pts.) Consider the following payoff matrix for a two person simultaneous move game: Column Player C C2 C3 R (4, 0) (-, 4) (,2) Row Player R2 (-, ) (2, 3) (2, 5) R3 (, ) (3, 3) (5,) Which of the following shows the equilibrium points (circled) and the reduced payoff matrix: C C2 C3 (a) R (4, 0) (-, 4) (,2) R2 (-, ) (2, 3) (2, 5) R3 (, ) (3, 3) (5,) Reduced C2 R3 (3, 3) (b) C C2 C3 R (4, 0) (-, 4) (,2) R2 (-, ) (2, 3) (2, 5) R3 (, ) (3, 3) (5,) Reduced C2 R3 (3, 3) (c) C C2 C3 R (4, 0) (-, 4) (,2) R2 (-, ) (2, 3) (2, 5) R3 (, ) (3, 3) (5,) Reduced C C2 C3 R (4, 0) (-, 4) (,2) R3 (, ) (3, 3) (5,) (d) C C2 C3 R (4, 0) (-, 4) (,2) R2 (-, ) (2, 3) (2, 5) R3 (, ) (3, 3) (5,) Reduced C C2 C3 R (4, 0) (-, 4) (,2) R3 (, ) (3, 3) (5,) (e) none of the above 4
5.(6 pts.) In the game of Rock-scissors-paper, the players face each other and simultaneously display their hands in one of the three following shapes: fist denoting a rock, the forefinger and middle finger extended and spread so as to suggest scissors, or downward facing palm denoting a sheet of paper. The rock wins over the scissors since it can shatter them, the scissors wins over the paper since they can cut it, and the paper wins over the rock since it can be wrapped around it. The winner collects a penny from the opponent and no money changes hands in the case of a tie. Raul and Carlita play a game of Rock-Scissors-Paper. Which of the matrices below represents the pay-off matrix for Raul, the row player, where R denotes rock, S denotes scissors and P denotes paper. (a) (b) (c) (d) (e) R S P R S P R S P R 0 S 0 P 0 R S P R 0 0 S 0 0 P 0 0 R S P R 0 S 0 P 0 R S P R 0 S 0 P 0 5
6.(6 pts.) The following graphs show the probability distributions of points scored per game played for Michael Jordan and Scottie Pippen. 0.0.02.03.04.05 0 0 20 30 40 50 Z, W Jordan Points Per Game = Z Pippen Points Per Game = W Which of the following statements is true? (a) (b) (c) (d) (e) The proportion of games played by Michael Jordan in which he scored more than 30 points was approximately equal to 0.04. The proportion of games played by Michael Jordan in which he scored more than 30 points was greater than 0.5. The average number of points scored per game for both players was the same. The proportion of games played by Scottie Pippen in which he scored less than 30 points was approximately equal to 0.02. The proportion of games played by Scottie Pippen played in which he scored more than 30 points was greater than 0.5. 6
7.(6 pts.) A ball is thrown directly upwards from a height of 4 meters above ground at a speed of 30 meters per second. What is the maximum height achieved by the ball? Recall that the force of gravity (on Earth) is approximately 9.8 meters per second. (a) Approximately 49.92 m. (b) Approximately 8 m. (c) Approximately 28.76 m. (d) Approximately 34 m. (e) Approximately 57.3 m. 7
Partial Credit You must show your work on the partial credit problems to receive credit! 8.(4 pts.) Two high school football players Robert and Chris can play effectively in a number of positions on the team. Robert who plays for the Eagles can play as running back (RB), quarterback(qb) or wide receiver(wr). Chris who plays for the Hawks can play tackle (T), middle linebacker (ML), or cornerback (C). Both teams are about to play in the regional championship game. The matrix below shows the estimated expected yardage gained per play for the Eagles (Robert s team) for each choice of positions for Robert and Chris. This can be considered as a zero sum game where yards gained by one team corresponds to yards lost by the other. Chris T ML C RB 5-5 8 Robert QB -5 5 20 WR 0 5 7 (a) Find the reduced payoff matrix by eliminating dominated strategies. (b) Find the optimal strategy for both players using the reduced matrix. (c) What is the expected pay-off for The Eagles (Robert s team) if both players play optimal strategies? 8
9.(2 pts.) On a reality TV show A contestant must complete a 2 step obstacle course to get to the next round. At the beginning of each stage of the obstacle course, the judge flips a fair coin to decide which task the contestant should complete. In the first stage of the obstacle course the contestant must either eat a bag of live worms or a bag of habanero peppers. If the contestant completes stage of the obstacle course, they get to attempt stage two, otherwise, they are out of the competition. In the second stage of the obstacle course the contestant must either retrieve a stick from a tank of alligators or a ball from a pit of snakes. If the contestant completes stage of the obstacle course, they get to move on to round two, otherwise, they are out of the competition. Sydney has a 50% chance of finishing a bag of worms and a 20% chance of finishing a bag of habaneros. She has a 30% chance of retrieving a stick from a tank of alligators(and living on to tell the tale) and a 20% chance of getting a ball out of a pit of snakes (without being poisoned). Use a tree diagram to determine the probability that Sydney will make it to the next round of the competition. 9
0.(2 pts.) The Great Zarini, the human cannonball is shot from a cannon at ground level over level terrain at an angle of 55 0 to the horizontal with an initial velocity of 20 meters per second. (a) Give a formula for The Great Zarini s velocity, t seconds after he was launched (v(t)). (b) What is the maximum height reached by The Great Zarini? (c) What was the horizontal distance covered by The Great Zarini in his flight? 0
.(20 pts.) Take Home Part
ANSWERS Math 070, Practice Exam 2 April 25, 205 The Honor Code is in effect for this examination. All work is to be your own. You may use your Calculator. The exam lasts for 50 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!. (a) (b) ( ) (d) (e) 2. (a) ( ) (c) (d) (e)... 3. (a) (b) (c) ( ) (e) 4. (a) ( ) (c) (d) (e)... 5. ( ) (b) (c) (d) (e) 6. (a) ( ) (c) (d) (e)... 7. ( ) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice 8. 9. 0.. Total