MGF 1106: Finite Math Review
MGF 1106 Finite Math Review # 2 E. Nicoli-Suco
-------------------Name___________________________________
*****In addition to this review, I recommend that you attend all classes, complete and review all homework assignments, and take any steps you consider necessary for yourself to succeed in the course.
Evaluate the expression. 1) 3! Answer: 6 2) 11C2 Answer: 55 3) 12C0 Answer: 1 4) 5P4 Answer: 120 5) 8P4 Answer: 1680 5) 4) 3) 2) 1)
6) How many different 4-digit sequences can be formed using the digits 0, 1,...,9 if repetition of digits is allowed? Answer: 10,000
6)
7) A shirt company has 3 designs each of which can be made with short or long sleeves. There are 5 color patterns available. How many different types of shirts are …show more content…
available from this company? Answer: 30
7)
1
8) How many different 8-digit sequences can be formed using the digits 0, 1,...,9 if repetition of digits is allowed? Answer: 100,000,000
8)
9) License plates are made using 2 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed? Answer: 676,000
9)
10) A shirt company has 3 designs each of which can be made with short or long sleeves. There are 6 color patterns available. How many different types of shirts are available from this company? Answer: 36
10)
11) If 4 newborn babies are randomly selected, how many different gender sequences are possible? Answer: 16
11)
12) How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed. Answer: 2997
12)
13) A restaurant offers 8 possible appetizers, 9 possible main courses, and 8 possible desserts. How many different meals are possible at this restaurant? (Two meals are considered different unless all three courses are the same). Answer: 576
13)
14) How many different 4-letter radio station call letters can be made if repeats are allowed and the first letter must be K. Answer: 17,576
14)
15) How many different 7-digit phone numbers are possible if the first digit cannot be a 0? Answer: 9,000,000
15)
2
16) How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, if repetition of digits is not allowed? Answer: 210
16)
17) How many ways can 6 people be chosen and arranged in a straight line if there are 8 people to choose from? Answer: 20,160
17)
18) A musician plans to perform 7 selections. In how many ways can she arrange the musical selections? Answer: 5040
18)
19) A musician plans to perform 4 selections. In how many ways can she arrange the musical selections? Answer: 24
19)
20) There are 9 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible? Answer: 504
20)
21) In a game of musical chairs, 10 children will sit in 9 chairs arranged in a row (one child will not find a chair). In how many ways can 9 of the children find seats? Answer: 3,628,800
21)
22) In how many ways can 2 letters be chosen from the set {A, B, C, D, E, F} if order is important and no repeats are allowed? Answer: 30
22)
23) There are 12 members on a board of directors. If they must form a subcommittee of 4 members, how many different subcommittees are possible? Answer: 495
23)
3
24) There are 13 members on a board of directors. If they must form a subcommittee of 6 members, how many different subcommittees are possible? Answer: 1716
24)
25) The library is to be given 5 books as a gift. The books will be selected from a list of 27 titles. If each book selected must have a different title, how many possible selections are there? Answer: 80,730 26) In a certain lottery, 4 numbers between 1 and 13 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important. Answer: 715
25)
26)
27) How many ways can an IRS auditor select 6 of 8 tax returns for an audit? Answer: 28
27)
28) Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female? Answer: 6
28)
29) Three student representatives, a president, a secretary, and a treasurer, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the president must be a woman and the secretary and treasurer must be men? Answer: 12
29)
30) Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex? Answer: 12
30)
31) A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committee can have at most two girls? Answer: 4620 ways
31)
4
A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability.
32) 2 cherry, 1 lemon Answer: .1818
32)
33) All lemon Answer: 0 34) All orange Answer: .0061
33)
34)
35) All cherry Answer: .1212
35)
36) One of each flavor Answer: .2182
36)
37) Six students, A, B, C, D, E, F, are to give speeches to the class. The order of speaking is determined by random selection. Find the probability that (a) E will speak first (b) that C will speak fifth and B will speak last (c) that the students will speak in the following order: DECABF (d) that A or B will speak first. Answer:
1 1 1 1 ; ; ; 6 24 720 3
37)
38) Amy, Jean, Keith, Tom, Susan, and Dave have all been invited to a birthday party. They arrive randomly and each person arrives at a different time. In how many ways can they arrive? In how many ways can Jean arrive first and Keith last? Find the probability that Jean will arrive first and Keith will arrive last. Answer: 720; 24;
1 30
38)
39) A group consists of 6 men and 5 women. Five people are selected to attend a conference. In how many ways can 5 people be selected from this group of 11? In how many ways can 5 men be selected from the 6 men? Find the probability that the selected group will consist of all men. Answer: 462; 6;
1 77
39)
5
40) A box contains 21 widgets, 4 of which are defective. If 4 are sold at random, find the probability that (a) all are defective (b) none are defective. Answer:
1 68 ; 5985 171
40)
41) A committee consisting of 6 people is to be selected from eight parents and four teachers. Find the probability of selecting three parents and three teachers. Answer:
8 33
41)
42) If you are dealt 5 cards from a shuffled deck of 52 cards, find the probability that all 5 cards are picture cards. Answer:
33 108290
42)
43) If you are dealt 6 cards from a shuffled deck of 52 cards, find the probability of getting 3 jacks and 3 aces. Answer:
2 2544815
43)
44) To play the lottery in a certain state, a person has to correctly select 5 out of 45 numbers, paying $1 for each five-number selection. If the five numbers picked are the same as the ones drawn by the lottery, an enormous sum of money is bestowed. What is the probability that a person with one combination of five numbers will win? What is the probability of winning if 100 different lottery tickets are purchased? Answer:
1 10 ; 1,221,759 1,221,759
44)
Find the requested probability. 45) A family has five children. The probability of having a girl is having no girls?
1 . What is the probability of 2
45)
Answer: .0313
A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting the given result. 46) Exactly one four Answer: .402 46)
6
In a certain college, 33% of the physics majors belong to ethnic minorities.
Find the probability of the event from a random sample of 10 students who are physics majors.
47) Exactly 2 belong to an ethnic minority. Answer: .1990
47)
Find the probability of the event. 48) A die is rolled 18 times and two threes come up. Answer: .230 48)
Solve the problem involving probabilities with independent events. 49) A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are colored green, and one is colored yellow. If the pointer is spun once, find the probability it will land on green and then yellow. Answer:
1 18
49)
50) A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are colored green, and one is colored yellow. If the pointer is spun three times, find the probability it will land on green every time. Answer:
1 27
50)
51) A single die is rolled twice. Find the probability of getting a 3 the first time and a 4 the second time. Answer:
1 36
51)
52) A single die is rolled twice. Find the probability of getting a 3 the first time and a 3 the second time. Answer:
1
36
52)
Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability.
53) Two balls are drawn from a bag in which there are 4 red balls and 2 blue balls. The number of blue balls is counted. Answer: x P 0 .4 1 .53 2 .07
53)
7
54) Four cards are drawn from a deck. The number of red tens is counted. Answer: x P 0 .851 1 .145 2 .005
54)
55) A ball player batting .300 comes to bat 3 times in a game. The number of hits is counted. Answer: x 0 1 2 3 P .343 .441 .189 .027
55)
56) Four coins are tossed and the number of heads is counted. Answer: x 0 1 2 3 4 P .0625 .25 .375 .25 .0625
56)
57) Three cards are drawn from a deck. The number of kings is counted. Answer: x 0 1 2 3 P .7826 .2042 .0130 .0002
57)
Find the expected value of the random variable in the experiment. 58) Three coins are tossed, and the number of tails is noted. Answer: 1.5 58)
59) Three cards are drawn from a deck without replacement. The number of aces is counted. Answer: .2308
59)
8
Find the expected value for the random variable. 60) y P(y) 6 8 10 12 0.4 0.4 0.13 0.07
60)
Answer: 7.73999991
61)
z
3
6
9
12
15
P(z) 0.14 0.14 0.36 0.26 0.10
61)
Answer: 9.12
62)
z
24
26
28
30
32
P(z) 0.23 0.12 0.46 0.17 0.08
62)
Answer: 29.18
63) A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day. Complaints per Day 0 1 2 3 4 5 Probability .04 .11 .26 .33 .19 .12 Answer: 2.98
63)
64) For a certain animal species, the probability that a female will have a certain number of offspring in a given year is given in the table below. Find the expected number of offspring per year. Number of Offspring 0 1 2 3 4 Probability .31 .21 .19 .17 .12 Answer: 1.58
64)
9
-------------------Name___________________________________
*****In addition to this review, I recommend that you attend all classes, complete and review all homework assignments, and take any steps you consider necessary for yourself to succeed in the course.
Evaluate the expression. 1) 3! Answer: 6 2) 11C2 Answer: 55 3) 12C0 Answer: 1 4) 5P4 Answer: 120 5) 8P4 Answer: 1680 5) 4) 3) 2) 1)
6) How many different 4-digit sequences can be formed using the digits 0, 1,...,9 if repetition of digits is allowed? Answer: 10,000
6)
7) A shirt company has 3 designs each of which can be made with short or long sleeves. There are 5 color patterns available. How many different types of shirts are …show more content…
available from this company? Answer: 30
7)
1
8) How many different 8-digit sequences can be formed using the digits 0, 1,...,9 if repetition of digits is allowed? Answer: 100,000,000
8)
9) License plates are made using 2 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed? Answer: 676,000
9)
10) A shirt company has 3 designs each of which can be made with short or long sleeves. There are 6 color patterns available. How many different types of shirts are available from this company? Answer: 36
10)
11) If 4 newborn babies are randomly selected, how many different gender sequences are possible? Answer: 16
11)
12) How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed. Answer: 2997
12)
13) A restaurant offers 8 possible appetizers, 9 possible main courses, and 8 possible desserts. How many different meals are possible at this restaurant? (Two meals are considered different unless all three courses are the same). Answer: 576
13)
14) How many different 4-letter radio station call letters can be made if repeats are allowed and the first letter must be K. Answer: 17,576
14)
15) How many different 7-digit phone numbers are possible if the first digit cannot be a 0? Answer: 9,000,000
15)
2
16) How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, if repetition of digits is not allowed? Answer: 210
16)
17) How many ways can 6 people be chosen and arranged in a straight line if there are 8 people to choose from? Answer: 20,160
17)
18) A musician plans to perform 7 selections. In how many ways can she arrange the musical selections? Answer: 5040
18)
19) A musician plans to perform 4 selections. In how many ways can she arrange the musical selections? Answer: 24
19)
20) There are 9 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible? Answer: 504
20)
21) In a game of musical chairs, 10 children will sit in 9 chairs arranged in a row (one child will not find a chair). In how many ways can 9 of the children find seats? Answer: 3,628,800
21)
22) In how many ways can 2 letters be chosen from the set {A, B, C, D, E, F} if order is important and no repeats are allowed? Answer: 30
22)
23) There are 12 members on a board of directors. If they must form a subcommittee of 4 members, how many different subcommittees are possible? Answer: 495
23)
3
24) There are 13 members on a board of directors. If they must form a subcommittee of 6 members, how many different subcommittees are possible? Answer: 1716
24)
25) The library is to be given 5 books as a gift. The books will be selected from a list of 27 titles. If each book selected must have a different title, how many possible selections are there? Answer: 80,730 26) In a certain lottery, 4 numbers between 1 and 13 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important. Answer: 715
25)
26)
27) How many ways can an IRS auditor select 6 of 8 tax returns for an audit? Answer: 28
27)
28) Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female? Answer: 6
28)
29) Three student representatives, a president, a secretary, and a treasurer, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the president must be a woman and the secretary and treasurer must be men? Answer: 12
29)
30) Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex? Answer: 12
30)
31) A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committee can have at most two girls? Answer: 4620 ways
31)
4
A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability.
32) 2 cherry, 1 lemon Answer: .1818
32)
33) All lemon Answer: 0 34) All orange Answer: .0061
33)
34)
35) All cherry Answer: .1212
35)
36) One of each flavor Answer: .2182
36)
37) Six students, A, B, C, D, E, F, are to give speeches to the class. The order of speaking is determined by random selection. Find the probability that (a) E will speak first (b) that C will speak fifth and B will speak last (c) that the students will speak in the following order: DECABF (d) that A or B will speak first. Answer:
1 1 1 1 ; ; ; 6 24 720 3
37)
38) Amy, Jean, Keith, Tom, Susan, and Dave have all been invited to a birthday party. They arrive randomly and each person arrives at a different time. In how many ways can they arrive? In how many ways can Jean arrive first and Keith last? Find the probability that Jean will arrive first and Keith will arrive last. Answer: 720; 24;
1 30
38)
39) A group consists of 6 men and 5 women. Five people are selected to attend a conference. In how many ways can 5 people be selected from this group of 11? In how many ways can 5 men be selected from the 6 men? Find the probability that the selected group will consist of all men. Answer: 462; 6;
1 77
39)
5
40) A box contains 21 widgets, 4 of which are defective. If 4 are sold at random, find the probability that (a) all are defective (b) none are defective. Answer:
1 68 ; 5985 171
40)
41) A committee consisting of 6 people is to be selected from eight parents and four teachers. Find the probability of selecting three parents and three teachers. Answer:
8 33
41)
42) If you are dealt 5 cards from a shuffled deck of 52 cards, find the probability that all 5 cards are picture cards. Answer:
33 108290
42)
43) If you are dealt 6 cards from a shuffled deck of 52 cards, find the probability of getting 3 jacks and 3 aces. Answer:
2 2544815
43)
44) To play the lottery in a certain state, a person has to correctly select 5 out of 45 numbers, paying $1 for each five-number selection. If the five numbers picked are the same as the ones drawn by the lottery, an enormous sum of money is bestowed. What is the probability that a person with one combination of five numbers will win? What is the probability of winning if 100 different lottery tickets are purchased? Answer:
1 10 ; 1,221,759 1,221,759
44)
Find the requested probability. 45) A family has five children. The probability of having a girl is having no girls?
1 . What is the probability of 2
45)
Answer: .0313
A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting the given result. 46) Exactly one four Answer: .402 46)
6
In a certain college, 33% of the physics majors belong to ethnic minorities.
Find the probability of the event from a random sample of 10 students who are physics majors.
47) Exactly 2 belong to an ethnic minority. Answer: .1990
47)
Find the probability of the event. 48) A die is rolled 18 times and two threes come up. Answer: .230 48)
Solve the problem involving probabilities with independent events. 49) A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are colored green, and one is colored yellow. If the pointer is spun once, find the probability it will land on green and then yellow. Answer:
1 18
49)
50) A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are colored green, and one is colored yellow. If the pointer is spun three times, find the probability it will land on green every time. Answer:
1 27
50)
51) A single die is rolled twice. Find the probability of getting a 3 the first time and a 4 the second time. Answer:
1 36
51)
52) A single die is rolled twice. Find the probability of getting a 3 the first time and a 3 the second time. Answer:
1
36
52)
Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability.
53) Two balls are drawn from a bag in which there are 4 red balls and 2 blue balls. The number of blue balls is counted. Answer: x P 0 .4 1 .53 2 .07
53)
7
54) Four cards are drawn from a deck. The number of red tens is counted. Answer: x P 0 .851 1 .145 2 .005
54)
55) A ball player batting .300 comes to bat 3 times in a game. The number of hits is counted. Answer: x 0 1 2 3 P .343 .441 .189 .027
55)
56) Four coins are tossed and the number of heads is counted. Answer: x 0 1 2 3 4 P .0625 .25 .375 .25 .0625
56)
57) Three cards are drawn from a deck. The number of kings is counted. Answer: x 0 1 2 3 P .7826 .2042 .0130 .0002
57)
Find the expected value of the random variable in the experiment. 58) Three coins are tossed, and the number of tails is noted. Answer: 1.5 58)
59) Three cards are drawn from a deck without replacement. The number of aces is counted. Answer: .2308
59)
8
Find the expected value for the random variable. 60) y P(y) 6 8 10 12 0.4 0.4 0.13 0.07
60)
Answer: 7.73999991
61)
z
3
6
9
12
15
P(z) 0.14 0.14 0.36 0.26 0.10
61)
Answer: 9.12
62)
z
24
26
28
30
32
P(z) 0.23 0.12 0.46 0.17 0.08
62)
Answer: 29.18
63) A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day. Complaints per Day 0 1 2 3 4 5 Probability .04 .11 .26 .33 .19 .12 Answer: 2.98
63)
64) For a certain animal species, the probability that a female will have a certain number of offspring in a given year is given in the table below. Find the expected number of offspring per year. Number of Offspring 0 1 2 3 4 Probability .31 .21 .19 .17 .12 Answer: 1.58
64)
9