Download - MATH104 Ch. 11: Probability Theory
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MATH104Ch. 11: Probability Theory
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Permutation Examples
1. If there are 4 people in the math club (Anne, Bob, Cindy, Dave), and we wish to elect a president and vice-president, LIST all of the different ways that this is possible.
2. From these 4 people (Anne, Bob, Cindy, Dave),
we wish to elect a president, vice-president, and treasurer. LIST all of the different ways that this is possible.
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Answers
1. If there are 4 people in the math club (Anne, Bob, Cindy, Dave), and we wish to elect a president and vice-president, LIST all of the different ways that this is possible.
AB BA CA DAAC BC CB DBAD BD CD DC
4*3=12 or 4P2 = 12
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Answers2. From these 4 people (Anne, Bob, Cindy, Dave),
we wish to elect a president, vice-president, and treasurer. LIST all of the different ways that this is possible.
ABCABD…
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• A B C ABC
D ABDC B ACB
D ACDD A BDA
C BDC• B A C BAC
D BCDC A BCA
D BCDD A BDA
C BDC• C A B CAB
D CADB A CBA
D CBDA B DAB
C DAC• D A B DAB
C DACB A DBA
C DBCC A DCA
B DCB
4*3*2 = 24 outcomesOr 4P3 = 24
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More counting examples:
1. At a restaurant, you have a choice of main dish (beef, chicken, fish, vegetarian), vegetable (broccoli, corn), potato (baked, fries), and dessert (chocolate, strawberry). LIST all possible choices.
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2. T/F quiz
2. A teacher wishes to make all possible different answer keys to a T/F quiz to cut down on cheating. How many possible different answer keys could there be if there are 4 questions. LIST them all.
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3. T/F test
3. What if there were 10 T/F questions. Just explain (do not list).
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4. Multiple choice test
4. A teacher wishes to make all possible different answer keys to a multiple choice quiz. How many possible different answer keys could there be if there are 3 questions that each have 4 choices (A,B,C,D). LIST them all.
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5. And 6.5. What if there were 20 multiple choice
questions with 5 choices each? Explain (don’t list).
6. With 9 baseball players on a team, how many different batting orders exist?
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Counting Rules Fundamental Counting/ –Multiplication Rule (p. 608) If you can choose one item from a group of M items and a
second item from a group of N items, then the total number of two-item choices is M*N.
Permutation of n things taken r at a time (p. 617) nPr = n!/(n-r)! Question: In permutations, does
ORDER matter? Is REPITITION allowed? Permutations of Duplicate items (p. 618) The number of permutations of n items, where p items
are identical, q items are identical, r items are identical, and so on, is given by
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More multiplication and permutation problems
1. With 14 players on a team, how many ways could we pick a batting order of 11?
2. If license plates have 3 letters and then 4
numbers, how many different license plates exist?
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3. A stock can go up, down, or stay unchanged. If you own 7 stocks, how many different possibilities are there?
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4. How many different four-letter radio station call letters can be formed if the first letter must be W or K?
5. A social security number contains nine
digits. How many different ones can be formed?
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6. If you wish to arrange your 7 favorite books on a shelf, how many different ways can this be done?
7. If you have 10 favorite books, but only have
room for 7 books on the shelf, how many ways can you arrange them?
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8. You wish to arrange 12 of your favorite photographs on a mantel. How many ways can this be done?
9. You have 20 favorite photographs and wish
to arrange 12 of them on a mantel. How many ways can that be done?
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10.
10. You take a multiple choice test with 12 questions (and each can be answered A B C D E). How many different ways could you answer the test?
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11. How many ways can you rearrange the letters in
a. CAT?
b. OHIO?
c. CLASSES?
d. MISSISSIPPI?
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12. If a station plans on running 6 (identical) Democratic ads, 6 (identical) Republican ads, and 4 (identical) Independent ads, in how many ways can they order these?
13. If you saw 15 movies last year, how many
ways can the top 3 be chosen and ranked?
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14. 20 people purchase raffle tickets. How many ways could we award a 1st, 2nd, and 3rd prize.
15. You have 50 different outfits. How many
ways can you pick your first and second favorite? How about your first, second, and third favorite?
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Combination Questions
1. If there are 4 people in the math club (Anne, Bob, Cindy, Dave), and 2 will be selected to attend the national math conference. LIST all of the different ways that this is possible.
2. From these 4 people (Anne, Bob, Cindy, Dave), and 3 will be selected to attend the national math conference. LIST all of the different ways that this is possible.
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Combination answers1. If there are 4 people in the math club (Anne,
Bob, Cindy, Dave), and 2 will be selected to attend the national math conference. LIST all of the different ways that this is possible.
ABAC BCAD BD CD
4C2= 6
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Combination answer
2. From these 4 people (Anne, Bob, Cindy, Dave), and 3 will be selected to attend the national math conference. LIST all of the different ways that this is possible.
ABC BCDABDACD
4C3 = 4
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Permutations and Combinations• Permutations– Use when ORDER matters and NO repitition– nPr = n!/(n-r)!– Example: If 10 people join a club, how many ways
could we pick pres and vp? 10P2 = 90• Combinations– Use: ORDER does NOT matter and NO repitition– nCr = n!/ [(n-r)!r!]– Example: 10 people join a club. In how many ways
could we pick 2? 10C2 = 45
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Combination of n things taken r at a time (p. 623)
Use the combination formula nCr = n!/[(n-r)!r!] to answer these combination problems
1. If there are 20 people on a committee, how many ways could we pick a subcommittee of 7 of them?
2 If there are 100 senators, how many ways could we pick a subcommittee of 7 of them?
3 If there are 72 potential jurors, how many different ways could they pick a jury of 12?
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Decide and answer: Combination, permutation, or multiplication?
1. There are 8 possible pizza toppings. How many ways could we pick 3 toppings?
2 . 20 people apply for a scholarship. 3 are chosen. In how many ways can they be chosen?
3. 32 people are in a class where the teacher plans on awarding 4 A’s. If all possibilities were written out, how many would there be?
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Change some of the following permutation problems into combination problems
1. Permutation question: With 14 players on a team, how many ways could we pick a batting order of 11? Answer: 14P11
Write a combination questions whose answer is 14C11
2. Permutation question: If you have 10 favorite
books, but only have room for 7 books on the shelf, how many ways can you arrange them?Answer: 10P7
Write a combination questions whose answer is 10C7
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…3. Permutation question: You have 20 favorite photographs
and wish to arrange 12 of them on a mantel. How many ways can that be done? Answer: 20P12
Write a combination questions whose answer is 20C12 4. Permutation question: If you saw 15 movies last year, how
many ways can the top 3 be chosen and ranked? Answer: 15P3
Write a combination questions whose answer is 15C3
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5. Permutation question: 20 people purchase raffle tickets. How many ways could we award a 1st, 2nd, and 3rd prize.Answer: 20P3
Write a combination questions whose answer is 20C3
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More challenging combination problems
1 If we have 4 teachers and 7 students and wish to form a committee of 2 teachers and 3 students, in how many different ways can this be done?
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2 . A test has 5 essay questions and 10 short answer questions. A student is to select to answer 3 essay questions and 7 short answers. In how many different ways could this be done?
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Multiplication Problems1. At a restaurant, you have a choice of main dish (beef, chicken, fish,
vegetarian), vegetable (broccoli, corn), potato (baked, fries), and dessert (chocolate, strawberry). LIST all possible choices.
2. A teacher wishes to make all possible different answer keys to a multiple
choice quiz. How many possible different answer keys could there be if there are 3 questions that each have 4 choices (A,B,C,D). LIST them all.
3. What if there were 20 multiple choice questions with 5 choices each?
Explain (don’t list). 4. With 9 baseball players on a team, how many different batting orders
exist?
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Answers1. At a restaurant, you have a choice of main dish (beef,
chicken, fish, vegetarian), vegetable (broccoli, corn), potato (baked, fries), and dessert (chocolate, strawberry). LIST all possible choices.
main vegetable potato dessert
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Answers
2. A teacher wishes to make all possible different answer keys to a multiple choice quiz. How many possible different answer keys could there be if there are 3 questions that each have 4 choices (A,B,C,D). LIST them all.
3. What if there were 20 multiple choice questions with 5
choices each? Explain (don’t list). 4. With 9 baseball players on a team, how many different
batting orders exist?
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Multiplication, Permutation, or Combination?
1. With 14 players on a team, how many ways could we pick a batting order of 11?
2. If license plates have 3 letters and then 4 numbers, how many different
license plates exist? 3. How many different four-letter radio station call letters can be formed if
the first letter must be W or K? 4. A social security number contains nine digits. How many different ones
can be formed? 5. If you wish to arrange your 7 favorite books on a shelf, how many
different ways can this be done?
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6. If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrange them?
7. You wish to arrange 12 of your favorite photographs on a mantel. How many ways can this be done?
8. You have 20 favorite photographs and wish to arrange 12 of them on a mantel. How many ways can that be done?
9. You take a multiple choice test with 12 questions (and each can be answered A B C D E). How many different ways could you answer the test?
10. If you had 13 pizza toppings, how many ways could you pick 5 of them?