MATH104 Ch. 11: Probability Theory

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. 

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 DA AC BC CB DB AD BD CD DC 4*3=12 or 4P2 = 12

Answers 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. ABC ABD…

4*3*2 = 24 outcomes Or 4P3 = 24 A B C ABC D ABD C B ACB D ACD D A BDA C BDC B A C BAC D BCD C A BCA C A B CAB D CAD B A CBA D CBD A B DAB C DAC D A B DAB B A DBA C DBC C A DCA B DCB 4*3*2 = 24 outcomes Or 4P3 = 24

More counting examples: 1.      At a restaurant, you have a choice of main dish (beef, chicken, fish), vegetable (broccoli, corn), potato (baked, fries), and dessert (chocolate, strawberry).  LIST all possible choices.  

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.

3. T/F test 3.      What if there were 10 T/F questions.  Just explain (do not list).

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 4 questions that each have 3 choices (A,B,C).  LIST all.

More multiplication problems 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?

More multiplication and permutation problems 1. With 14 players on a team, how many ways could we pick a batting order of 11? 2. From 12 possible speakers, how many ways could you select an order of 5?

More multiplication and permutation 3. With 8 new movies available, how many ways could you select 3 to watch one weekend (where order matters)?  4. If license plates have 3 letters and then 4 numbers, how many different license plates exist?

More Q 5.      How many different four-letter radio station call letters can be formed if the first letter must be W or K?   6.      A social security number contains nine digits.  How many different ones can be formed?

… 7. If you wish to arrange your 7 favorite books on a shelf, how many different ways can this be done? 8. If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrange them?

… 9. You wish to arrange 12 of your favorite photographs on a mantel. How many ways can this be done? 10. You have 20 favorite photographs and wish to arrange 12 of them on a mantel. How many ways can that be done?

… 11.  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?

Counting Rules- Summary 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)!               In permutations, ORDER matter & REPETITION is NOT 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  

10. How many ways can you rearrange the letters in a.  CAT? BOB? OHIO?  d. CLASSES?  e. MISSISSIPPI?

A few more examples—which rule? 11. 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? 12. If you saw 15 movies last year, how many ways can the 1st, 2nd, and 3rd by chosen?

13 13.  20 people purchase raffle tickets.  How many ways could we award a 1st, 2nd, and 3rd prize.    14.  You have 50 different outfits.  How many ways can you pick your first and second favorite?

Combination Questions 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. 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.

Combination answers 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. AB AC BC AD BD CD 4C2= 6

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 BCD ABD ACD 4C3 = 4

Permutations and Combinations Use when ORDER matters and NO repetition 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 repetition nCr = n!/ [(n-r)!r!] Example: 10 people join a club. In how many ways could we pick 2? 10C2 = 45

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? 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? .

Decide and answer: Combination, permutation, or multiplication? There are 8 possible pizza toppings.  How many ways could we pick 3 toppings?  2 . 20 people apply for a $1000 scholarship.  3 are chosen.  In how many ways can they be chosen?  3. There are 8 colors of pants in the store. How many ways could we choose to buy 3 different colors?

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

…rewrite the problem 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

5. Permutation question: 20 people purchase raffle tickets 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

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?  

… 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?

Review -- Multiplication, Permutation, or Combination? 1. If we have 13 people in class, how many ways could we schedule 13 oral reports? 2. Although we have 13 people, we only have time for 10 reports today. In how many ways could we do this?    3.     A password contains 8 characters (which can be letters or digits). With no other restrictions, how many different ones can be formed? 4. There are 8 colors of sweaters at the mall. How many ways could we select 3 to buy?