AP STATS REVIEW PROBABILITY (Ch

AP STATS REVIEW – PROBABILITY

(2002B #2)

Airlines routinely overbook flights because they expect a certain number of no-shows. An airline runs a 5 P.M. commuter flight from Washington, D.C. to New York City on a plane that holds 38 passengers. Past experience has shown that if 41 tickets are sold for the flight, then the probability distribution for the number who actually show up for the flight is as shown in the table below.

Number who actually show up / 36 / 37 / 38 / 39 / 40 / 41
Probability / .46 / .30 / .16 / .05 / .02 / .01

Assumer 41 tickets are sold for each flight.

a)There are 38 passenger seats on the flight. What is the probability that all passengers who show up for this flight will get a seat?

b)What is the expected number of no-shows for this flight?

c)Given that not all passenger seats are filled on a flight, what is the probability that only 36 passengers showed up for the flight?

(2003B #2)

A simple random sample of adults living in a suburb of a large city was selected. The age and annual income of each adult in the sample were recorded. The resulting data are summarized in the table below.

Annual Income
Age Category / $25,000-$35,000 / $35,001-$50,000 / Over $50,000 / Total
21-30 / 8 / 15 / 27 / 50
31-45 / 22 / 32 / 35 / 89
46-60 / 12 / 14 / 27 / 53
Over 60 / 5 / 3 / 7 / 15
Total / 47 / 64 / 96 / 207

What is the probability that a person chosen at random from those in this sample will be in the 31-45 age category?

What is the probability that a person chosen at random from those in this sample whose incomes are over $50,000 will be in the 31-45 age category? Show your work.

Based on your answers to parts (a) and (b), is annual income independent of age category for those in this sample? Explain.

(2003B #5)

Contestants on a game show spin a wheel like the one shown in the figure to the right. Each of the for outcomes on this wheel is equally likely and outcomes are independents from one spin to the next.

The contestant spins the wheel.
If the result is a skunk, no money is won and the contestant’s turn is finished.
If the result is a number, the corresponding amount in dollars in won. The contestant con then stop with those winnings or can choose to spin again, and his or her turn continues.

What is the probability that the result will be a number on all of the first three spins of the wheel?

Suppose a contestant has earned $800 on his or her first three spins and chooses to spin the wheel again. What is the expected value of his or her total winnings for the four spins?

A contestant who lost at this game alleges that the wheel is not fair. In order to check on the fairness of the wheel, the data in the table below were collected for 100 spins of this wheel.

Result / Skunk / $100 / $200 / $500
Frequency / 33 / 21 / 20 / 26

Based on these data, can you conclude that the four outcomes on this wheel are not equally likely? Give appropriate statistical evidence to support your answer.

(2006 #3)

The depth from the surface of Earth to a refracting layer beneath the surface can be estimated using methods developed by seismologists. One method is based on the time required for vibrations to travel from a distantexplosion to a receiving point. The depth measurement (M) is the sum of the true depth (D) and the randommeasurement error (E). That is, M = D + E. The measurement error (E) is assumed to be normally distributedwith mean 0 feet and standard deviation 1.5 feet.

a)If the true depth at a certain point is 2 feet, what is the probability that the depth measurement will be negative?

b)Suppose three independent depth measurements are taken at the point where the true depth is 2 feet. What isthe probability that at least one of these measurements will be negative?

c)What is the probability that the mean of the three independent depth measurements taken at the point wherethe true depth is 2 feet will be negative?

(2006B #3)

Golf balls must meet a set of five standards in order to be used in professional tournaments. One of these standards is distance traveled. When a ball is hit by a mechanical device, Iron Byron, with a 10-degree angle oflaunch, a backspin of 42 revolutions per second, and a ball velocity of 235 feet per second, the distance the balltravels may not exceed 291.2 yards. Manufacturers want to develop balls that will travel as close to the 291.2yards as possible without exceeding that distance. A particular manufacturer has determined that the distancestraveled for the balls it produces are normally distributed with a standard deviation of 2.8 yards. Thismanufacturer has a new process that allows it to set the mean distance the ball will travel.

a)If the manufacturer sets the mean distance traveled to be equal to 288 yards, what is the probability that aball that is randomly selected for testing will travel too far?

b)Assume the mean distance traveled is 288 yards and that five balls are independently tested. What is theprobability that at least one of the five balls will exceed the maximum distance of 291.2 yards?

c)If the manufacturer wants to be 99 percent certain that a randomly selected ball will not exceed themaximum distance of 291.2 yards, what is the largest mean that can be used in the manufacturing process?

(2007B #2)

The graph below displays the relative frequency distribution for X, the total number of dogs and cats owned per household, for the households in a large suburban area. For instance, 14 percent of the households own 2 ofthese pets.

a)According to a local law, each household in this area is prohibited from owning more than 3 of these pets. Ifa household in this area is selected at random, what is the probability that the selected household will be inviolation of this law? Show your work.

b)If 10 households in this area are selected at random, what is the probability that exactly 2 of them will be inviolation of this law? Show your work.

c)The mean and standard deviation of X are 1.65 and 1.851, respectively. Suppose 150 households in this areaare to be selected at random and X , the mean number of dogs and cats per household, is to be computed.Describe the sampling distribution of X , including its shape, center, and spread.

(2008B #5)

Flooding has washed out one of the tracks of the Snake Gulch Railroad. The railroad has two parallel tracks from Bullsnake to Copperhead, but only one usable track from Copperhead to Diamondback, as shown in the figurebelow. Having only one usable track disrupts the usual schedule. Until it is repaired, the washed-out track willremain unusable. If the train leaving Bullsnake arrives at Copperhead first, it has to wait until the train leavingDiamondback arrives at Copperhead.

Every day at noon a train leaves Bullsnake heading for Diamondback and another leaves Diamondback heading for Bullsnake.

Assume that the length of time, X, it takes the train leaving Bullsnake to get to Copperhead is normallydistributed with a mean of 170 minutes and a standard deviation of 20 minutes.

Assume that the length of time, Y, it takes the train leaving Diamondback to get to Copperhead is normallydistributed with a mean of 200 minutes and a standard deviation of 10 minutes.

These two travel times are independent.

a)What is the distribution of Y -X ?

b)Over the long run, what proportion of the days will the train from Bullsnake have to wait at Copperhead forthe train from Diamondback to arrive?

c)How long should the Snake Gulch Railroad delay the departure of the train from Bullsnake so that theprobability that it has to wait is only 0.01 ?

MULTIPLE CHOICE:

CHAPTER 6

1.An assignment of probability must obey which of the following?

(a)The probability of any event must be a number between 0 and 1, inclusive.

(b)The sum of all the probabilities of all outcomes in the sample space must be exactly 1.

(c)The probability of an event is the sum of the outcomes in the sample space which make up the event.

(d)All of the above.

(e)A and B only.

2.Event A occurs with probability 0.2. Event B occurs with probability 0.8. If A and B are disjoint (mutually exclusive), then

(a)P(A and B) = 0.16.

(b)P(A or B) = 1.0.

(c)P(A and B) = 1.0.

(d)P(A or B) = 0.16.

(e)Both A and B are true.

3.A fair coin is tossed four times, and each time the coin lands heads up. If the coin is then tossed 1996 more times, how many heads are most likely to appear for these 1996 additional tosses?

(a)996

(b)998

(d)1996

(e)None of the above. The answer is ______.

4.A die is loaded so that the number 6 comes up three times as often as any other number. What is the probability of rolling a 1 or a 6?

(a)1/3

(b)1/4

(d)2/3

(e)None of the above. The answer is ______.

Questions 5 and 6 relate to the following: In a particular game, a fair die is tossed. If the number of spots showing is either four or five, you win $1. If the number of spots showing is six, you win $4. And if the number of spots showing is one, two, or three, you win nothing. You are going to play the game twice.

5.The probability that you win $4 both times is

(a)1/6

(b)1/3

(d)1/4

(e)1/12

6.The probability that you win at least $1 both times is

(a)1/2

(b)4/36

(d)1/4

(e)3/4

Question 7 and 8 relate to the following: An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.1.

7.The conditional probability of A given B

(a)is 0.5.

(b) is 0.3.

(c)is .2.

(d)is 1/6.

(e)cannot be determined from the information given.

8.We may conclude that

(a)events A and B are independent.

(b)events A and B are disjoint.

(c)either A or B always occurs.

(d)events A and B are complementary.

(e)none of the above.

9.Experience has shown that a certain lie detector will show a positive reading (indicates a lie) 10% of the time when a person is telling the truth and 95% of the time when a person is lying. Suppose that a random sample of 5 suspects is subjected to a lie detector test regarding a recent one-person crime. Then the probability of observing no positive reading if all suspects plead innocent and are telling the truth is

(a)0.409

(b)0.735

(d)0.591

(e) 0.99999

10.If you buy one ticket in the Provincial Lottery, then the probability that you will win a prize is 0.11. If you buy one ticket each month for five months, what is the probability that you will win at least one prize?

(a)0.55

(b)0.50

(d) 0.45

(e)0.56

CHAPTER 7

A psychologist studied the number of puzzles subjects were able to solve in a five-minute period while listening to soothing music. Let X be the number of puzzles completed successfully by a subject. X had the following distribution:

X1234

Probability 0.2 0.4 0.3 0.1

1.Using the above data, what is the probability that a randomly chosen subject completes at least 3 puzzles in the five-minute period while listening to soothing music?

(a)0.3

(b)0.4

(d)0.9

(e)The answer cannot be computed from the information given.

2.Using the above data, P(X < 3) is

(a)0.3

(b)0.4

(d)0.9

(e)The answer cannot be computed from the information given.

3.Using the above data, the mean µ of X is

(a)2.0

(b)2.3

(d)3.0

(e)The answer cannot be computed from the information given.

4. Which of the following random variables should be considered continuous?

(a)The time it takes for a randomly chosen woman to run 100 meters

(b)The number of brothers a randomly chosen person has

(c)The number of cars owned by a randomly chosen adult male

(d)The number of orders received by a mail order company in a randomly chosen week

(e)None of the above

5.Let the random variable X represent the profit made on a randomly selected day by a certain store.

Assume that X is normal with mean $360 and standard deviation $50. What is the value of P(X >$400)?

(a)0.2119

(b)0.2881

(d)0.8450

6.A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000 profit. If it is very cold that day, she expects to suffer a $12,000 loss. Based upon historical records, the weather office has estimated the chances of a warm day to be .60; the chances of a cool day to be .25. What is the producer's expected profit?

(a)$5,000

(b)$13,000

(c)$15,050

(d)$13,250

(e)$11,450

7.In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5 you win $1, if number of spots showing is 6 you win $4, and if the number of spots showing is 1, 2, or 3 you win nothing. Let X be the amount that you win. The expected value of X is

(a)$0.00.

(b)$1.00.

(c)$2.50.

(d)$4.00.

(e)$6.00.

Questions 6 and 7 use the following: Suppose X is a random variable with mean µX and standard deviation X. Suppose Y is a random variable with mean µY and standard deviation Y.

8.The mean of X + Y is

(a)µX+ µY .

(b)(µX / X) + (µY / Y).

(c)µX+ µY, but only if X and Y are independent.

(d)(µX/ X) + (µY/ Y), but only if X and Y are independent.

(e)None of these.

9.The variance of X + Y is

(a)X + Y.

(b)(X)2 + (Y)2.

(c)X + Y, but only if X and Y are independent.

(d)(X)2 + (Y)2, but only if X and Y are independent.

10.Suppose X is a continuous random variable taking values between 0 and 2 and having the probability density function below.

P(1 ≤ X ≤ 2) has value

(a)0.50.

(b)0.33

(d)0.00

(e)None of these.

CHAPTER 8

1.It has been estimated that about 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. A consumer purchases 12 frozen chickens. What is the probability that the consumer will have more than 6 contaminated chickens?

(a)0.961

(b)0.118

(d) 0.039

(e)0.079

2.The probability that a certain machine will produce a defective item is 0.20. If a random sample of 6 items is taken from the output of this machine, what is the probability that there will be 5 or more defectives in the sample?

(a)0.0001

(b)0.0154

(d)0.2458

(e)0.0016

3.A professional basketball player sinks 80% of his foul shots, in the long run. If he gets 100 tries during a season, then the probability that he sinks between 75 and 90 shots (inclusive) is approximately equal to:

(a)Pr ( -1.25 <= Z <= 2.5 )

(b)Pr ( -1.125 <= Z <= 2.625 )

(c)Pr (-1.125 <= Z <= 2.375 )

(d)Pr ( -1.375 <= Z <= 2.375 )

(e)Pr (-1.375 <= Z <= 2.625 )

4.If X has a binomial distribution with n = 400 and p = .4, the approximate probability of the event {155 < X < 175} is:

(a)0.6552

(b)0.6429

(d)0.6201

(e)0.6320

5.If in the previous question we change the interval to 155 <= X <= 175, the approximate probability is;

(a)0.4

(b)Larger than that in the previous question

(c)Smaller than that in the previous question

(d)Equal to that in the previous question

(e)May be smaller or larger than that in the previous question

6.A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let X denote the number in the sample that say they support the increase. Suppose that 40% of all adults in Ohio support the increase. The probability that X is more than 650 is

(a) less than 0.0001.

(b) less than 0.001.

(d) 0.9960.

(e) none of these.

7.A fair coin (one for which both the probability of heads and the probability of tails are 0.5) is tossed six times. Use the binomial formula to evaluate the probability that less than 1/3 of the tosses are heads is

(a) 0.344.

(b) 0.33.

(d) 0.09.

(e) 0.0043.

8.Suppose we select an SRS of size n = 100 from a large population having proportion p of successes. Let X be the number of successes in the sample. For which value of p would it be safe to assume the sampling distribution of X is approximately normal?

(a) 0.01

(b) 1/9

(d) 0.9999

(e) All of these.

9.Suppose we roll a fair die ten times. The probability that an even number occurs exactly the same number of times as an odd number on the ten rolls is

(a) 0.1667.

(b) 0.2461.

(d) 0.5000.

(e) None of these.

10.In a large population of college students, 20% of the students have experienced feelings of math anxiety. If you take a random sample of 10 students from this population, the probability that exactly 2 students have experienced math anxiety is

(a) 0.3020

(b) 0.2634

(d) 0.5

(e)1

(f)None of the above

11.Refer to the previous problem. The standard deviation of the number of students in the sample who have experienced math anxiety is

(a)0.0160

(b)1.265

(d)1

(e) .2070

CHAPTER 9

1.A phone-in poll conducted by a newspaper reported that 73% of those who called in liked business tycoon Donald Trump. The unknown true percentage of American citizens who like Donald Trump is a

(a)Statistic

(b)Sample

(c)Parameter

(d)Population