AP Statistics – Unit 4 (Chapters 14-16) Warm-Ups: Part 1

1.	I toss a penny and observe whether it lands heads up or tails up. Suppose the penny is fair, that is, the probability of heads is ½ and the probability of tails is ½. This means 	
A)	that every occurrence of a head must be balanced by a tail in one of the next two or three tosses	
B)	that if I flip the coin many, many times, the proportion of heads will be approximately ½, and this proportion will tend to get closer and closer to ½ as the number of tosses increases	
C)	that regardless of the number of flips, half will be heads and half tails      D) all of the above	

2.	A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. This proportion represents 	
A)	the probability of the event A                B) the distribution of the event A	
C)	the correlation of the event A                D) the variance of the event A	

3.	A basketball player makes 160 out of 200 free throws. We would estimate the probability that the player makes his next free throw to be 	
A)	0.16             B) 50-50; either he makes it or he doesn't              C) 0.80             D) 1.2	

6.	Suppose we have a loaded die that gives the outcomes 1–6 according to the following probability distribution:
 X	
1	
2	
3	
4	
5	
6	

P(X)	
0.1	
0.2	
0.3	
0.2	
0.1	
0.1	

Note that for this die all outcomes are not equally likely, as they would be if the die were fair. If this die is rolled 6000 times, the number of times we get a 2 or a 3 should be about 	
A)	1000                B) 2000                 C) 3000                D) 4000	

7.	Suppose we roll a red die and a green die. Let A be the event that the number of spots showing on the red die is 3 or less and B be the event that the number of spots showing on the green die is more than 3. Events A and B are 	
A)	disjoint            B) complements             C) independent             D) reciprocals	

9.	I select two cards from a deck of 52 cards and observe the color of each (26 cards in the deck are red and 26 are black). Which of the following is an appropriate sample space S for the possible outcomes? 	
A)	S = {red, black} 	
B)	S = {(red, red), (red, black), (black, red), (black, black)}, where, for example, (red, red) stands for the event “the first card is red and the second card is red”     C) S = {0, 1, 2}      D) all of the above	
 
If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. Assume the table below gives the probability that a randomly chosen M&M has each color.
Color	Brown	Red	Yellow	Green	Orange	Tan	
Probability	.3	.3	?	.1	.1	.1 	
11.	The probability of drawing a yellow candy is 	
A)	.1              B) .2            C) .3           D) impossible to determine from the information given
	
12.	The probability that I do not draw a red candy is    A) .1        B) .3      C) .7       D) .9	

13.	The probability that I draw neither a brown nor a green candy is    A) .1       B) .3       C) .4      D) .6	

17.	Event A occurs with probability 0.2. Event B occurs with probability 0.8. If A and B are disjoint 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	

19.	Suppose there are three cards in a deck, one marked with a 1, one marked with a 2, and one marked with a 5. You draw two cards at random and without replacement from the deck of three cards. The sample space S = {(1, 2), (1, 5), (2, 5)} consists of these three equally likely outcomes. Let X be the total of the two cards drawn. Which of the following is the correct set of probabilities for X? 	
A)	X	
 1	
 2	
 5                              B) 	
X	
3	
6	
 7	

P(X)	
1/3	
1/3	
1/3	
P(X)	
1/3	
1/3	
1/3	

C)	X	
3	
6	
 7                              D)	
 X	
3	
6	
 7	

P(X)	
3/16	
6/16	
7/16	
P(X)	
1/4	
1/2	
1/4	

A stack of four cards contains two red cards and two black cards. I select two cards, one at a time, and do not replace the first card selected before selecting the second card. 
20.	Referring to the information above, consider the events
A = the first card selected is red       B = the second card selected is red        The events A and B are 	
A)	independent              B) disjoint               C) complements             D) none of the above	

21.	Referring to the information above, consider the events
A = the first card selected is red            B = the second card selected is black              The events A and B are 	
A)	independent              B) disjoint               C) complements             D) none of the above	

25.	Suppose that A and B are two independent events with P(A) = 0.3 and P(B) = 0.3.  P(A È B) is 	
A)	0.09                B) 0.51                  C) 0.52                  D) 0.60	

26.	Suppose that A and B are two independent events with P(A) = .2 and P(B) = .4.  P(A Ç BC) is 	
A)	0.08               B) 0.12               C) 0.52                 D) 0.60	

A system has two components that operate in parallel, as shown in the diagram below. Since the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probabilities of failures for the components 1 and 2 during one period of operation are .20 and .03, respectively. Let F denote the event that component 1 fails during one period of operation and G denote the event that component 2 fails during one period of operation. The component failures are independent. 

28.	The event corresponding to the system failing during one period of operation is 	
A)	F Ç G           B) F È G           C) Fc È Gc             D) Fc Ç Gc	

29.	The event corresponding to the system functioning properly during one period of operation is 	
A)	F Ç G              B) F È G              C) Fc È Gc                   D) Fc Ç Gc	

30.	The probability that the system functions properly during one period of operation is closest to 	
A)	0.994              B) 0.970            C) 0.940               D) 0.776	

31.	The probability that the system fails during one period of operation is closest to 	
A)	0.230              B) 0.224               C) 0.060              D) 0.006	

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. 
32.	Referring to the information above, the conditional probability of A given B 	
A)	is 0.3              B) is 0.2             C) is 1/6            D) cannot be determined from the information given	

33.	Referring to the information above, we may conclude 	
A)	that events A and B are independent                B) that events A and B are disjoint	
C)	that either A or B always occurs                       D) none of the above	

38.	The probability of a randomly selected adult having a rare disease for which a diagnostic test has been developed is 0.001. The diagnostic test is not perfect. The probability the test will be positive (indicating that the person has the disease) is 0.99 for a person with the disease and 0.02 for a person without the disease. The proportion of adults for which the test would be positive is 	
A)	0.00099                    B) 0.01998                C) 0.02097                 D) 0.02100	

39.	In question 38, if a randomly selected person is tested and the result is positive, the probability the individual has the disease is 	
A)	0.019                 B) 0.020              C) 0.021               D) 0.047	

AP Statistics – Unit 4 (Chapters 14-16) Warm-Ups: Part 2

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. The psychologist found that X had the following probability distribution:
Value of X	1	2	3	4	
Probability	0.2	0.4	0.3	0.1	

1.	Referring to the information above, the probability that a randomly chosen subject completes at least three puzzles in the five-minute period while listening to soothing music is 	
A)	0.3              B) 0.4                 C) 0.6               D) 0.9	

2.	Referring to the information above, P(X < 3) has value 	
A)	0.3              B) 0.4                 C) 0.6               D) 0.9	

The probability density of a random variable X is given in the figure at the right
7.	Referring to the information above, from this density, 
the probability that X is between 0.5 and 1.5 is 	
A)	1/3             B) 1/2              C) 3/4                D) 1	

8.	Referring to the information above, the probability that X is at least 1.5 is 	
A)	0                B) 1/4                 C) 1/3                   D) 1/2         	

9.	Referring to the information above, the probability that X = 1.5 is 	
A)	0                B) 1/4                 C) 1/3                   D) 1/2         	

11.	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               C) 0.25                  D) 0.00	

Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume X is normal with a mean of $360 and standard deviation $50. 
13.	Referring to the information above, the value of P(X > $400) is 	
A)	0.2881                B) 0.8450                 C) 0.7881                   D) 0.2119	

14.	Referring to the information above, the probability is approximately 0.6 that on a randomly selected day the store will make less than 	
A)	$347.40                    B) $0.30                  C) $361.30                     D) $372.60	

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. 
15.	Referring to the information above, the expected value of X is 	
A)	$0             B) $1              C) $2.50             D) $4	

16.	Referring to the information above, the variance of X is 	
A)	1.0              B) 3/2               C) 2.0               D) 13/6	

The weight of medium-size tomatoes selected at random from a bin at the local supermarket is a random variable with 
mean  = 10 ounces and standard deviation  =  1 ounce. 
20.	Suppose we pick four tomatoes from the bin at random and put them in a bag. The weight of the bag is a random variable with a mean of 	
A)	2.5 ounces               B) 4 ounces                C) 10 ounces               D) 40 ounces	

21.	Suppose we pick four tomatoes from the bin at random and put them in a bag. The weight of the bag is a random variable with a standard deviation (in ounces) of 	
A)	0.25                 B) 0.50                 C) 2                 D) 4	

22.	The weight of the tomatoes in pounds (1 pound = 16 ounces) is a random variable with standard deviation 	
A)	1/16 pounds              B) 1 pound               C) 16 pounds                D) 256 pounds	

23.	Suppose we pick two tomatoes at random from the bin. The difference in the weights of the two tomatoes selected (the weight of first tomato minus the weight of the second tomato) is a random variable with a standard deviation (in ounces) of                 A) 0                   B) 1                  C) 1.41                 D) 2	

27.	I toss a fair coin a large number of times. Assuming tosses are independent, which of the following is true? 	
A)	Once the number of flips is large enough (usually about 10,000) the number of heads will always be exactly half of the total number of tosses. For example, after 10,000 tosses I should have 5000 heads	
B)	The proportion of heads will be about ½, and this proportion will tend to get closer and closer to ½ as the number of tosses increases	
C)	As the number of tosses increases, any long run of heads will be balanced by a corresponding run of tails so that the overall proportion of heads is ½            D) all of the above	

A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability distribution.
X	0	1	2	3	4	
P(X)	0.1	0.1	0.1	0.1	0.6	

28.	Referring to the information above, the mean number of customers that make a purchase during the first hour that the store is open is 	
A)	2                B) 2.5             C) 3                D) 4	

29.	Referring to the information above, the standard deviation of the number of customers that make a purchase during the first hour that the store is open is 	
A)	1.4             B) 2                C) 3                D) 4	

30.	Referring to the information above, suppose the store is open seven days a week from 8:00 AM to 5:30 PM. The mean number of customers that make a purchase during the first hour that the store is open during a one-week period is 	
A)	3                B) 9             C) 21            D) 28	

34.	Suppose we have a loaded die that gives the outcomes 1–6 according to the probability distribution 
 X	
1	
2	
3	
4	
5	
6	

P(X)	
0.1	
0.2	
0.3	
0.2	
0.1	
0.1	

Note that for this die all outcomes are not equally likely, as they would be if this die were fair. If this die is rolled 6000 times, then J, the sample mean of the number of spots on the 6000 rolls, should be about 	
A)	3              B) 3.30             C) 3.50              D) 4.50	

Answer Key (part I)
1.B	2.A	3.C	6.C	7.C	9.B	11.A	12.C	13.D	17.B	19.B	20.D
21.D	25.B	26.B	28.A	29.C	30.A	31.D	32.C	33.C	38.C	39.D

Answer Key (part II)
1.B	2.C	7.B	8.B	9.A	11.C	13.D	14.D	15.B	16.C	20.D	21.C	22.A
23.C	27.B	28.C	29.A	30.C	34.B