Great Ideas in Mathematics
Math 110 - Fall 2007
Ch. 15: Probability and odds
Practice questions
More questions will be posted later
Some answers are below.
1. A coin is tossed three times. Which of the following describes the sample space for this random experiment?
A {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
B {HHH, TTT}
C {H, T}
D {3 H’s, 2 H’s and 1 T, 1 H and 2 T’s, 3 T’s}
E none of the above
2. If an honest coin is tossed three times, the probability of tossing 3 heads is
A 1/9
B 2/9
C 1/3
D 2/3
E none of these
3. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many different passwords are there?
A 365.
B 536.
C 36 x 5
D 265 + 105
E none of the above
4. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many do not start with the digit 0?
A 2/27
B 1/3
C 1/27
D 1/9
E none of the above
5. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many start with a digit?
A 8 / 38.
B 1 / 36.
C 7 / 83.
D 7 / 38.
E None of these
6. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many consist entirely of letters?
A 265.
B 26 x 25 x 24 x 23 x 22
C 26 x 5
D 526.
E none of the above
7. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many have four letters and only one digit?
A 26 x 25 x 24 x 23 x 10
B 4 x 26 x 10
C 5 x 264 x 10
D 264 x 10
E none of the above
8. A computer password is made up of five characters. Each character can be a capital letter (A through Z) or a digit (0 through 9). How many have 3 letters and 2 digits?
A 10 x 263 x 102
B 263 x 102
C 3 x 263 x 102
D 26 x 25 x 24 x 10 x 9
E none of the above
9. Four basketball teams called A, B, C and D are entered in a tournament. According to the odds makers, the probability that team A will win the tournament is Pr(A) = 0.1, and the other three teams all have equal probabilities of winning the tournament. What is the probability that A will not win the tournament?
A 0.3
B 0.4
C 0.9
D It cannot be determined from the given information
E none of the above
10. Four basketball teams called A, B, C and D are entered in a tournament. According to the odds makers, the probability that team A will win the tournament is Pr(A) = 0.1, and the other three teams all have equal probabilities of winning the tournament. What is the probability that D will win the tournament?
A 0.25
B 0.3
C 0.45
D 0.9
E none of the above
11. A couple is planning to have four children. For this family, the probability of a boy is 46% and for a girl is 54%. (Assume that the gender of each child is independent from that of the other children.) What is the probability that they will have four boys?
A 0.464.
B 0.46 + 0.46 + 0.46 + 0.46
C 1/4
D 0.463 x 0.54
E All of the above are random experiments.
12. A couple is planning to have four children. For this family, the probability of a boy is 46% and for a girl is 54%. (Assume that the gender of each child is independent from that of the other children.) What is the probability that they will have one girl and three boys?
A 4 x 0.463 x 0.54
B 0.463 x 0.54
C 0.54 + 0.46 + 0.46 + 0.46
D 1/4
E All of the above are random experiments.
13. A couple is planning to have four children. For this family, the probability of a boy is 46% and for a girl is 54%. (Assume that the gender of each child is independent from that of the other children.) What is the probability that they will have an equal number of girls and boys?
A 4 x 0.462 x 0.542.
B 6 x 0.462 x 0.542.
C 0.462 x 0.542.
D 0.462 + 0.542.
E All of the above are random experiments.
14. Which of the following is not a random experiment?
A tossing a coin
B predicting the winner of next year’s World Series
C drawing a card from a deck of cards
D rolling a pair of dice
E All of the above are random experiments
15. In a general probability model, which of the following statements is not true?
A The impossible event always has probability equal to 0.
B All probabilities are equal.
C The probability of the sample space is always equal to 1.
D All probabilities are between 0 and 1 (0 and 1 included.)
E All of the above statements are true.
The
next five questions refer to the spinner shown below.
Each of the three regions have 120°
angles.
16. Suppose we spin this spinner twice, and on each spin, note on which region it stops. The sample space for this random experiment is
A {A, B, C}
B {AA, AB, AC, BA, BB, BC, CA, CB, CC}
C {A, A, B, B, C, C}
D {AA, AB, AC, BB, BC, CC}
E None of the above
17. If we spin the spinner twice, the probability that it will land on ‘A’ both times is
A 1/3
B 1/9
C 2/3
D 2/9
E None of the above.
18. If the spinner is spun six times and the region where the spinner stops is noted after each spin, how many different outcomes are there in the sample space?
A 36.
B 3 x 6
C 6
D 63.
E None of the above.
19. Suppose the spinner is spun three times. What is the probability of spinning 2 Bs and 1 A?
A 2/27
B 1/27
C 1/9
D 1/3
E None of the above.
20. Suppose the spinner is spun eight times, What is the probability of spinning at least 7 As.
A 7/83.
B 1/36.
C 8/38.
D 7/38.
E None of the above.
The next three questions refer to the following example: A license plate consists of 6 symbols: three numerals 1-9 and three capital letters from the ordinary English alphabet (A through Z) except for the letters O, I, and Q.
21. How many different license plates are possible?
A 9 x 8 x 7 x 23 x 22 x 21
B 9 x 3 x 23 x 3
C 93 x 233.
D 9 x 23
E none of the above
22. How many of the license plates have no repeated letters or digits.
A 9 x 8 x 7 x 23 x 22 x 21
B 9 x 3 x 23 x 3
C 93 x 233.
D 9 x 23
E none of the above
23. How many of the license plates end with the word ADD ?
A 93 x 23
B 9 x 8 x 7
C 9 x 3
D 93.
E None of the above
24. 9P2 =
A 36
B 72
C 181,440
D 18
E none of the above
25. A Tasmanian lottery ticket consists of choosing six different numbers from 10 to 54. The number of possible lottery tickets is given by
A (54 !) / (48 ! x 6 !)
B (45 ! ) / (6 ! x 39 !)
C (45 !) / (29 !)
D 46 x 6
E None of the above
26. 110 golfers start a tournament. Assuming all of the golfers are equally skilled and that there are no ties, how many top 5 finishes are possible?
A 110C5.
B 110 x 5
C 105!
D 110P5.
E none of the above
27. If the chances of rain tomorrow are 37.5%, then the odds of rain tomorrow can be given as
A 3 to 11
B 37.5 to 1
C 3 to 7
D 3 to 8
E none of the above
The next two refer to the following example: A catering service offers a menu consisting of 6 different choices of fish, 5 different salads, 2 different soups, 4 different side dishes, and 3 different desserts.
28. A standard wedding meal is defined by the catering service as a choice of fish, a choice of soup or salad, a side dish, and a dessert. How many different standard wedding meals are possible?
A 720
B 20
C 504
D 168
E None of the above
29. A ‘special saver’ wedding meal is defined by the catering service as a choice of fish, a choice of soup, a side dish, and cake. How many different ‘special saver’ wedding meals are possible?
A 120
B 12
C 48
D 49
E None of the above
30. Three cards are drawn in order from a well-shuffled deck of 52 cards. The probability that all three cards are face cards (a face card is a Jack, a Queen, or a King) is given by
A
B
C
D
E None of the above
31. Suppose that three unfair coins are tossed simultaneously, and we count the numbers of heads and tails. The probabilities of the outcomes in the sample space are given in the table below:
|
X = number of heads |
0 |
1 |
2 |
3 |
|
Pr(X) |
27/64 |
27/64 |
9/64 |
1/64 |
Using these probabilities for the outcomes, match the probabilities of the events given below with their probabilities.
|
The probability that all heads occur |
A |
63/64 |
|
|
The probability that exactly two heads occur |
B |
10/64 |
|
|
The probability that at least one head occurs |
C |
54/64 |
|
|
The probability that at most two heads occur |
D |
37/64 |
|
|
The probability that at least two heads occur |
E |
1/64 |
|
|
The probability that less than two heads occur |
F |
9/64 |
32. A pair of unfair six-sided dice are tossed. The probabilities of the outcomes in the sample space are given in the table below:
|
X |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|
Pr(X) |
1/36 |
2/36 |
6/36 |
5/36 |
3/36 |
2/36 |
3/36 |
5/36 |
6/36 |
2/36 |
1/36 |
Using these probabilities for the outcomes, match the probabilities of the events given below with their probabilities.
|
The probability that a seven does not occur |
A |
3/36 |
|
|
The probability that at most a three occurs |
B |
8/36 |
|
|
The probability that at least a four occurs |
C |
4/36 |
|
|
The probability that a two, three or twelve occurs |
D |
33/36 |
|
|
The probability that a seven or eleven occurs |
E |
3/36 |
|
|
The probability that at least an eleven occurs |
F |
34/36 |
33. Baskin-Robbins features 31 different flavors of ice cream available in a cup, a sugar cone or a waffle cone. Match the given ice cream order to the total number of ways in the list.
|
The number of ways to order one dip in a cup container |
A |
C(31,1) = 31 |
|
|
The number of ways to order two dips in a sugar cone |
B |
C(31,2)=465 |
|
|
The number of ways to order three dips in a waffle cone |
C |
C(2,1)*C(31,2)=930 |
|
|
The number of ways to order two dips in one of two types of containers |
D |
C(3,1)*C(31,1)=93 |
|
|
The number of ways to order three dips in one of three types of containers |
E |
C(3,1)*C(31,3)=13485 |
|
|
The number of ways to order one dip in any one of three types of containers |
F |
C(31,3)=4495 |
34. What is special about the way Persi Diaconis shuffles cards?
35. What is special about the way Persi Diaconis flips coins?
Updated November 11, 2007
Some of the Answers
1 A
2 E
3 A
4
5
6 A
7 C
8 A
9 C
10 B
11 A
12 A
13 B
14 B
15 B
16 B
17 B
18 A
19 C
20 C
21 C
22 A
23 D
24 B
25 B
26 D
27 E
28 C
29 C
30 A
31 EFDABC
32 FADCCE
33 ABFCED
End of the answers