4.11. In a game, you draw thirteen cards with replacement from a deck of playing cards. If you draw any
y aces or twos, you lose the game immediately. You also lose if you draw picture cards(J,Q,K) more than twice. In this question, you’ll study the probability of winning this game.(a) What is the probability of drawing no aces or twos after thirteen draws?(b) Given you have drawn thirteen times, none of which is aces or twos, what is the probability that you draw at most two picture cards?(c) What is the probability to win this game?
12. Suppose you are tossing an unbiased coin for100times.(a) What is the probability of getting50heads and50tails?(b) LetXbe the random variable counting the number of heads you observe in this exper-iment. What is the expected value ofX? What is the variance ofX? What is thestandard deviation ofX?
13. The following are probability distributions for two random variablesX,Y.
(a) Construct the probability distribution table for the random variableXY.(b) Find E[X],E[Y] and E[XY]. Is is true that E[XY] =E[X]E[Y]?(c) Find the variances σ2X,σ2Y,σ2XY of X,Y and XY. Is it true that σ2XY=σ2Xσ2Y?
14. The aliens who are fond of gambling came back to play another game with you. In this game, you first toss a coin5times. If you observe3or fewer tails, you roll a die3times. If youobserve4or more tails, you roll a die20times. What is the probability that you end up with at most two6’s in your dice rolls?
15. (Challenge question, worth2points) You have two bags, each of which contains10marbles.Each time you remove a marble from a random bag. What is the probability that after one of the bags is emptied, there are still exactly3marbles in the other bag?
6.Exercise 4) A fair coin is tossed. If it lands heads, a fair four-sided die
is thrown (with values 2,3,4,7). If
2,3,4,7). If it lands tails, a fair six-sided die is
thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice
eats n grains of rice, where n is the largest prime factor of the die result
(for example, the largest prime factor of 9 is 3).
(a) What is the conditional probability that the coin lands heads, given
that Alice eats three grains of rice?
(b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through).
What is the conditional probability that the coin lands heads on both
run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs?
(Do not count the two grains from part (a) in part (b); we assume
two brand new experiments, each with a new coin toss. Start your
solution by defining a suitable partition of the sample space. Please
use an appropriate notation and/or justification in words, for each
value that you give as part of your solution.)
Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 .
Each time the unfair coin lands heads, Alice gives Bob £1 . Each time
the unfair coin lands tails, Bob gives Alice £1 . The game ends when one
player has £5 .
(a) Draw a labelled Markov chain describing the problem, and write
down a transition matrix P. Write down the communication classes,
and classify them as either recurrent or transient.
(b) Using the transition matrix, calculate the probability that Alice loses
all of her money in exactly four tosses of the unfair coin.
(c) Calculate the (total) probability that Alice loses all of her money
(before Bob loses all of his).
(d) Calculate the expected (mean) number of tosses of the unfair coin,
for the game to end.
7.Can domestic dogs understand human body cues such as leaning? The experimenter leaned toward one of two objects and recorded
two objects and recorded whether or not the dog being tested correctly chose the object indicated. A four-year-old male beagle named Augie participated in this study. He chose the correct object 42 out of 70 times when the experimenter leaned towards the correct object.
(a) (2 points) Let the parameter of interest, π, represent the probability that the long-run probability that Augie chooses correctly. Researches are interested to see if Augie understands human body cues (better than gussing).
Fill in the blanks for the null and alternative hypotheses.
H0 : Ha :
(b) (6 points) Based on the above context, conduct a test of significance to determine the p-value to investigate if domestic dogs understand human body cues. What conclusion will you draw with significance level of 10%? (If you use an applet, please specify which applet you use, and the inputs.)
(c) (5 points) Based on the above context, conduct a test of significance to determine the p-value to investigate if domestic dogs understand human body cues. What conclusion will you draw with significance level of 5%? (If you use an applet, please specify which applet you use, and the inputs.)
(d) (2 points) Are your conclusions from part (b) and (c) the same? If they are different, please provide an explanation.
(e) (5 points) Shown below is a dotplot from a simulation of 100 sample proportions under the assump- tion that the long-run probability that Augie chooses correct is 0.50. Based on this dotplot, would a 90% confidence interval for π contain the value 0.5? Explain your answer.
(f) (4 points) Compute the standard error of the sample proportion of times that Augie chose the object correctly.
(g) (5 points)
(h) (3 points) question?
(i) (4 points)
(j) (4 points) A.
Construct an approximate 95% confidence interval for π using the 2SD method. What is the margin of error of the confidence interval that you found in the previous
How would you interpret the confidence interval that you found in part (g)?
Which of the following is a correct interpretation of the 95% confidence level?
If Augie repeats this process many times, then about 95% of the intervals produced will capture the true proportion of times of choosing the correct objective.
About 95% times Augie chooses the correct objective.
If Augie repeats this process and constructs 20 intervals from separate independent sam- ples, we can expect about 19 of those intervals to contain the true proportion Augie chooses the correct objective.
(k) (4 points)
object 21 out of 35 times.
Conjecture how, if at all, the center and the width of a 99% confidence interval would change with these data, compared to the original 2SD 95% confidence interval.
The center of the confidence interval would . The width of the confidence interval would .
(l) (4 points) Suppose that we repeated the same study with Augie, and this time he chose the correct object 17 out of 35 times, and we also change the confidence level from 95% to 99%. Conjecture how, if at all, the center and the width of a 99% confidence interval would change with these data, compared to the original 2SD 95% confidence interval.
Suppose that we repeated the same study with Augie, and this time he chose the correct
The center of the confidence interval would The width of the confidence interval would
8.Question 1: What is a player’s « reaction function » in a Bertrand game ?
Question 2: What is a subgame
subgame perfect Nash equilibrium?
Question 3: In which situations should we need the mixed extension of a game?
Question 4: Find, if any, all Nash equilibria of the following famous matrix game:
U (2,0) (3,3)
D (3,4) (1,2)
Question 5: What is the difference between a separating equilibrium and a pooling equilibrium
in Bayesian games?
Question 6: Give another name for, if it exists, the intersection of the players’ best-response
« functions » in a game?
Question 7: assuming we only deal with pure strategies, the Prisoner’s Dilemma is a situation
No Nash equilibrium One sub-optimal Nash equilibrium
One sub-optimal dominant profile No dominant profile
Question 8: If it exists, a pure Nash equilibrium is always a profile of dominant strategies:
Question 9: All games have at least one pure strategy Nash equilibrium:
Question 10: If a tree game has a backward induction equilibrium then it must also be a Nash
equilibrium of all of its subgames:
Question 11: The mixed Nash equilibrium payoffs are always strictly smaller than the pure
Nash equilibrium payoffs:
Question 12: Which of the following statements about dominant/dominated strategies is/are
I. A dominant strategy dominates a dominated strategy in 2x2 games.
II. A dominated strategy must be dominated by a dominant strategy in all games.
III. A profile of dominant strategies must be a pure strategy Nash equilibrium.
IV. A dominated strategy must be dominated by a dominant strategy in 2x2 games.
I, II and IV only I, II and III only II and III only
I and IV only I, III and IV only I and II only
Question 13: A pure strategy Nash equilibrium is a special case of a mixed strategy Nash
Question 14: Consider the following 2x2 matrix game:
U (3,2) (2,4)
D (-1,4) (4,3)
The number of pure and mixed Nash equilibria in the above game is:
Exercise (corresponding to questions 15 to 20 below): assume a medical doctor (M)
prescribes either drug A or drug B to a patient (P), who complies (C) or not (NC) with each of
this treatment. In case of compliance, controlled by an authority in charge of health services
quality, the physician is rewarded at a level of 1 for drug A and 2 for drug B. In case of noncompliance, the physician is « punished » at -1 level for non-compliance of the patient with
drug A and at -2 level for non-compliance with drug B. As for the compliant patient, drug A
should give him back 2 years of life saved and drug B, only 1 year of life saved. When noncompliant with drug A, the same patient wins 3 years of life (due to avoiding unexpected
allergic shock for instance), and when non-compliant with drug B, the patient loses 3 years of
Question 15: You will draw the corresponding matrix of the simultaneous doctor-patient game.
Question 16: Find, if any, the profile(s) of dominant strategies of this game.
Question 17: Find, if any, the pure strategy Nash equilibrium/equilibria of this game.
Question 18: Find, if any, the mixed strategy Nash equilibrium/equilibria of this game.
Questions 19 and 20: Now the doctor prescribes first, then the patient complies or not: draw
the corresponding extensive-form game (= question 19) AND find the subgame perfect Nash
9.Directions: You are part of a fireworks crew assembling a local fireworks display.
There are two parts to the fireworks platforms:
rts to the fireworks platforms: one part is on the ground and the
other part is on top of a building. You are going to graph all of your results on one
coordinate plane. Make sure to label each graph with its equation. Use the following
equations to assist with this assignment.
• The function for objects dropped from a height where t is the time in
seconds, h is the height in feet at time it t, and 0 h is the initial height is
0 ht t h ( ) 16 =− + .
• The function for objects that are launched where t is the time in seconds, h is
the height in feet at time t, 0 h is the initial height, and 0 v is the initial velocity
in feet per second is 2
0 0 ht t vt h ( ) 16 =− + + .
Select the link below to access centimeter grid paper for your portfolio.
Centimeter Grid Paper
First, conduct some research to help you with later portions of this portfolio
• Find a local building and estimate its height. How tall do you think the
• Use the Internet to find some initial velocities for different types of fireworks.
What are some of the initial velocities that you found?
Respond to the following items.
1. While setting up a fireworks display, you have a tool at the top of the
building and need to drop it to a coworker below.
a. How long will it take the tool to fall to the ground? (Hint: use the first
equation that you were given above, 2
0 ht t h ( ) 16 =− + . For the building’s
height, use the height of the building that you estimated in Task 1.)
b. Draw a graph that represents the path of this tool falling to the
ground. Be sure to label your axes with a title and a scale. Your graph
should show the height of the tool, h, after t seconds have passed.
Label this line “Tool”.
10.XYP (Ltd) embarks on a project to construct the townhouse in gauteng. They have requested you to draft a budget
ft a budget proposal in order for them to go and seek funding. The management team supplies you with the following information.
The company will not generate any income for the first 3 months of the year (jan - march) on the 4th month the company will generate sales of R500 000.00 thereafter the company sales will increase by 10% until December.
Salaries will be R85 000.00 a month from the beginning of the year. There will be increase of 7% which will only be applicable as if the sixth month. Staff bonus will be payable in December and will amount R28 000
Stationary printing expenses are estimated to be R10 000. 00( vat excluded) every month.
Telephone expenses in January are expected to be R5 500.00(vat excluded) and will increase by 5% every month.
Rent will be payable from the 3rd month at a cost of R 20 000.00. There will be decrease of 5% from August to the end if the year and the company will be given a rental discount of 10% in December.
Return on the investment will be as follows: april R90 000.00 August and September will be R80 000.00 december R150 000.00 all returns will be taxable according appropriate tax standards.
Draw a cash budget for XYP (pty) Ltd for the year ending 31 December 2018 (30)
14.I am given the following scenario:
The process of interference is assumed to be responsible for much of forgetfulness in human
uch of forgetfulness in human memory. The processing of new information interferes with old memories. One demonstration of interference examines the process of forgetting while subjects are asleep versus while they are awake. Because there should be less interference during sleep, there also should be less forgetting. The following data are from an experiment examining six groups of participants. All participants were given a group of words to remember. Then half of the participants went to sleep, and the others stayed awake (factor A). Within both the asleep and awake groups, one third of the participants were tested after 2 hours, one third after 4 hours, and one third after 8 hours (factor B). The experimenter recorded the number of words correctly recalled.
We were given a bunch of SPSS post hoc output tables (attached). We need to draw conclusions (APA statements) from the post hoc and simple effects. I do not understand what information I am looking for.