Reference no: EM13489073

Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard. Suppose that P(A) = .5, P(B) = .4, and P(A intersect B) = .25.
Question 1. Compute the probability that the selected student has at least one of the two types of cards.

Incorrect: Your answer is incorrect.
Question 2. What is the probability that the selected student has neither type of credit card?

Question 3.Calculate the probability that the selected student has a Visa card but not a MasterCard, that is, calculate P(A intersect B') (Draw a picture)


2.-/4 points My Notes
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The option to buy extended warranties is commonplace with most electronic purchases. But does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table.
Purchased Warranty?
Yes No Total
Item Purchased Digital Camera 30 42 72
Laptop Computer 145 203 348
Total 175 245 420
Question 1. What is the probability that a consumer purchased a digital camera and an extended warranty? (Express your answer as a decimal between 0 and 1, not a percent between 0 and 100).
(use 2 decimal places in your answer)
Question 2. What is the probability that a consumer purchased a laptop computer and an extended warranty? (Express your answer as a decimal between 0 and 1, not a percent between 0 and 100).
(use 2 decimal places in your answer)
Question 3. What is the probability that a consumer purchased an extended warranty given that he or she purchased a digital camera? (Express your answer as a decimal between 0 and 1, not a percent between 0 and 100)
(use 3 decimal places in your answer).
Question 4. What is the probability that a consumer purchased an extended warranty given that he or she purchased a laptop computer? (Express your answer as a decimal between 0 and 1, not a percent between 0 and 100)
(use 3 decimal places in your answer).
3.-/4.5 points My Notes
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A Probability-based Assessment of Stealing 2nd Base
stealing_secondAn often-used strategy in baseball is for a base runner on first base to steal second base. Historically the probability of a successful steal of second base is approximately 0.68, thus the probability of an unsuccessful steal is 0.32 (we'll ignore the relatively rare event in which the catcher throws the ball into center field allowing the base runner to advance to third base).
The table below lists the probabilities of scoring at least one run in situations that are defined by the number of outs and the bases occupied. These probabilities are determined from the analysis of thousands of games and game situations in the American League. For example, the probability of scoring at least one run when there are no outs and a runner on first base .39.

Probability of Scoring at Least One Run
Bases Empty 1st Base 2nd Base 3rd Base 1st, 2nd 1st, 3rd 2nd, 3rd 1st, 2nd, 3rd
0 Out .26 .39 .57 .72 .59 .76 .83 .81
1 Out .16 .26 .42 .55 .45 .61 .74 .67
2 Out .07 .13 .24 .28 .24 .37 .37 .43

Question. Suppose there is a runner on first base; second base and third base are unoccupied. For each of the possible number of outs (0, 1, and 2) determine the probability of scoring at least one run before an attempted steal, after an attempted steal, and indicate whether or not it is a good strategy for the runner to attempt to steal second base.
0 outs
probability of scoring at least one run before the attempted steal
probability of scoring at least one run after the attempted steal (use 2 decimal places)
It is a good strategy for the runner to attempt to steal second base.
True
False

1 out
probability of scoring at least one run before the attempted steal
probability of scoring at least one run after the attempted steal (use 2 decimal places)
It is a good strategy for the runner to attempt to steal second base.
True
False

2 outs
probability of scoring at least one run before the attempted steal
probability of scoring at least one run after the attempted steal (use 2 decimal places)
It is a good strategy for the runner to attempt to steal second base.
True
False
4.-/3.5 points My Notes
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On the first Friday of the 2011 NCAA Basketball tournament, in the West region number 1 seed Duke played number 16 seed Hampton and number 8 seed Michigan played number 9 seed Tennessee. The winners of these 2 games then played each other on Sunday. Before the Friday games were played, what was the probability that Duke would play Michigan on Sunday? In other words, what was the pregame probability that Duke won its Friday game AND Michigan won its Friday game?

A formula that provides an excellent prediction for the probability that a team will beat an opponent is the log 5 rule, which works as follows. Suppose team P is playing team Q; team P has won proportion p of its games and team Q has won proportion q of its games. The probability that team P beats team Q is estimated to be

(p - p*q)/(p + q - 2p*q)
This formula has several desirable properties (you may want to verify these properties to better understand the formula):
The probability P beats Q plus the probability Q beats P equals 1.
If p = q then the probability P beats Q is 0.50.
If p = 1 (that is P is undefeated) and q is not equal to 1, then P always beats Q.
If p = 0 (that is P is winless) and q is not equal to 0, then P always loses to Q.
If p > 1/2 and q < 1/2 then the probability P beats Q is greater than p.
If q = 1/2 then the probability P wins is p (and similarly if p = 1/2 then Q wins with probability q).

Even better estimates are obtained from the log 5 rule using values of p and q derived from the Pythagorean Theorem for Basketball, which estimates the proportion of games won by a basketball team with an expression involving the number of points a team scores and the number of points that team allows other teams to score. Specifically,
(text(points scored))^11.5/((text(points scored))^11.5 + text((points allowed))^11.5) ~= proportion of games team wins.

The table below shows points scored and points allowed during the 2010-2011 season.

Points Scored Points Allowed
Duke 2756 2185
Hampton 2152 1983
Michigan 2182 2069
Tennessee 2331 2221

Question. Use the information in the table and the log 5 rule to find the probability that Duke wins on Friday, the probability that Michigan wins on Friday, and the probability that Duke and Michigan play each other on Sunday (that is, that Duke AND Michigan win on Friday). (Assume games are independent).
probability that Duke wins (use 3 decimal places)
probability that Michigan wins (use 3 decimal places)
probability that Duke AND Michigan win (use 3 decimal places).
5.-/4.5 points My Notes
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Suppose 1 in 25 adults is afflicted with a disease for which a new diagnostic test has been developed. Given that an individual actually has the disease, a positive test result will occur with probability .99. Given that an individual does not have the disease, a negative test result will occur with probability .98.
Use a probability tree to answer the following questions.
Question 1. What is the probability of a positive test result?

Question 2. If a randomly selected adult is tested and the result is positive, what is the probability that the individual has the disease?

Question 3. If a randomly selected adult is tested and the result is negative, what is the probability that the individual does not have the disease?

6.-/3.5 points My Notes
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The sport of football does not allow high school players to play professional football until 3 years after high school graduation. Therefore essentially all professional football players first play college football.
Historically, 5.7% of high school football players go on to play at the college level. Of these, only 1.8% will play in the National Football League (NFL). The average career length in the NFL is about 3.5 seasons according to the National Football League Players Association. 40% of NFL players have a career of more than 3 seasons.
Question 1. Use the probability tree approach to determine the probability that a high school football player competes in college and then plays in the NFL.
Use 5 decimal places in your answer.
Question 2. Use the probability tree approach to determine the probability that a high school football player competes in college and then goes on to have an NFL career of more than 3 seasons.
Use 5 decimal places in your answer.
NOTE: The economic appeal of being a professional football player is unquestionably strong, but hopes of fortune should be tempered by the statistical fact that NFL players with careers longer than 3 seasons have an average life expectancy that is 20 years less than that of the typical U.S. male.
7.-/3 points My Notes
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After graduation your first job is with Abibas Inc., a company that specializes in sportswear with collegiate logos. To improve profits the company wants to introduce a new selection of sportswear. The company will choose either the UNC-CH or NC STATE logo.

The marketing department's research shows that the odds are 96 to 1 that the new sportswear based on the UNC-CH logo will be profitable, while the odds are only 24 to 1 that the new sportswear based on the NC STATE logo will be profitable. Therefore the marketing department concludes that the UNC-CH logo is the overwhelming favorite.

The marketing department also says that since 96 = 4*24, the company should spend 4 times as much money developing the UNC-CH sportswear than it would spend developing the NC STATE sportswear.

You claim that the situation is not as overwhelming as it appears and that spending that much more on the UNC-CH logo would be a mistake.

Question 1. Calculate the probability that the new sportswear based on the UNC-CH logo will be profitable.

Question 2. Calculate the probability that the new sportswear based on the NCSTATE logo will be profitable.

8.-/1 points My Notes
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If there are 9 baseball players that need to be arranged in a batting order, how many different batting orders are possible?

9.-/2 points My Notes
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basketball startersSuppose that a basketball team has 12 players. Five of these players are classified as guards and 7 of the players are classified as forwards/centers.
Question 1. The coach writes each player's name on a piece of paper and places the 12 pieces of paper in a hat. If for a particular game the coach selects the 5 starting players by randomly selecting names from the hat, how many different groups of 5 starting players are possible?

Question 2. To avoid an unbalanced starting lineup, the coach wants to choose 2 guards and 3 forwards/centers. To do this the coach places the names of the 5 guards in one hat and the names of the 7 forwards/centers in a second hat. Then the coach selects 2 guards from the first hat and 3 forwards/centers from the second hat. How many starting lineups are possible?