1. 6/4/2016MATH 106, Section 191 Section 20 Dice Last Section we will cover!!!! We will finish this section in this class or early next class. Use the remainder of the class time to work on the homework assignments Quiz #6 COMING UP! Be sure to do the review problems for this, quiz posted on the internet. The link can be found in the course schedule.

2. 6/4/2016MATH 106, Section 192 Dice of many different colors are available. Each one is a standard, fair, six-sided die. (a) The red die and the blue die are rolled. Use each of the three methods outlined as follows to find the probability that at least one side facing upward is a 3. U = E = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} {(1,3) (2,3) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,3) (5,3) (6,3)} #U = #E = METHOD #1 36 11 #E P(E) = —– = #U 11 — = 0.306 or 30.6% 36

3. 6/4/2016MATH 106, Section 193 U = A = B = E = {all pairings of the integers 1 to 6, repetition possible} {all pairings of 3 followed by each of the integers 1 to 6} METHOD #2 (a) The red die and the blue die are rolled. Use each of the three methods outlined below to find the probability that at least one side facing upward is a 3. #U = #E = #E P(E) = —– = #U 6 2 = 36 A  BA  B {all pairings of each of the integers 1 to 6 followed by 3} #(A  B) = #A + #B – #(A  B) = 6 +6 –1 = 11 — = 0.306 or 30.6% 36

4. 6/4/2016MATH 106, Section 194 U = E = ~E = {all pairings of the integers 1 to 6, repetition possible} {all pairings of the integers 1 to 6 with at least one 3, repetition possible} METHOD #3 (a) The red die and the blue die are rolled. Use each of the three methods outlined below to find the probability that at least one side facing upward is a 3. #U = #E = #~E = #E P(E) = —– = #U {all pairings of the integers 1 to 6 with no 3s, repetition possible} 6 2 = 36 5 2 = 25 36 – 25 = 11 11 — = 0.306 or 30.6% 36 Which method would be the easiest to use, if three or more dice are rolled instead of just two dice?

5. 6/4/2016MATH 106, Section 195 (b) The red die, the blue die, the yellow die, the purple die, and the brown die are rolled. Find the probability that at least one side facing upward is a 3. U = E = ~E = {all 5-tuples of the integers 1 to 6, repetition possible} {all 5-tuples of the integers 1 to 6 with at least one 3, repetition possible} #U = #E = #~E = #E P(E) = —– = #U {all 5-tuples of the integers 1 to 6 with no 3s, repetition possible} 6 5 = 7776 5 5 = 3125 7776 – 3125 = 4651 4651 —— = 0.598 or 59.8% 7776

6. 6/4/2016MATH 106, Section 196 (c) The red die, the blue die, the yellow die, the purple die, the brown die, and the green die are rolled. Find the probability that at least one side facing upward is a 3. U = E = ~E = {all 6-tuples of the integers 1 to 6, repetition possible} {all 6-tuples of the integers 1 to 6 with at least one 3, repetition possible} #U = #E = #~E = #E P(E) = —– = #U {all 6-tuples of the integers 1 to 6 with no 3s, repetition possible} 6 6 = 46,656 5 6 = 15,625 46,656 –15,625 = 31,031 31,031 ——— = 0.665 or 66.5% 46,656

7. 6/4/2016MATH 106, Section 197 (d) Ten dice of different colors are rolled. Find the probability that no side facing upward is a 3. U = E = {all 10-tuples of the integers 1 to 6, repetition possible} #U = #E = #E P(E) = —– = #U {all 10-tuples of the integers 1 to 6 with no 3s, repetition possible} 6 10 = 60,466,176 5 10 = 9,765,625 9,765,625 ————— = 0.1615 or 16.15% 60,466,176

8. 6/4/2016MATH 106, Section 198 Homework Hints: In Section 20 Homework Problems #1 and #2, the approach that can be used is any of the approaches used in parts (a), (b), (c), and (d) of #1 on the Section #20 Handout.

9. 6/4/2016MATH 106, Section 199 (e) The red die and the blue die are rolled. Use each of the two methods outlined as follows to find the probability that the sum of the sides facing upward is 9. U = E = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} {(3,6) (4,5) (5,4) (6,3)} #U = #E = METHOD #1 36 4 #E P(E) = —– = #U 4 — = 0.111 or 11.1% 36

10. 6/4/2016MATH 106, Section 1910 U = E = {all pairings of the integers 1 to 6, repetition possible} {all integer solutions to r + b = 9 with 1  r  6 AND 1  b  6} METHOD #2 (e) The red die and the blue die are rolled. Use each of the two methods outlined as follows to find the probability that the sum of the sides facing upward is 9. #U = #E = 6 2 = 36 #E P(E) = —– = #U r + b = 7 r  5 AND b  5 Use GOOD = ALL – BAD r + b = 7 6  r OR 6  b 2! 2! —— + —— – 0 = 4 1! 1! 1! 1! 8! —— – 4 = 4 1! 7! 4 4 — = 0.111 or 11.1% 36 Which method would be the easiest to use, if three or more dice are rolled instead of just two dice?

11. 6/4/2016MATH 106, Section 1911 (f) The red die, the blue die, the yellow die, the purple die, and the brown die are rolled. Find the probability that the sum of the sides facing upward is 15. U = E = {all 5-tuples of the integers 1 to 6, repetition possible} {all integer solutions to r + b + y + p + w = 15 with 1  r  6 AND 1  b  6 AND 1  y  6 AND 1  p  6 AND 1  w  6} r + b + y + p + w = 10 r  5 AND b  5 AND y  5 AND p  5 AND w  5 Use GOOD = ALL – BAD r + b + y + p + w = 10 6  r OR 6  b OR 6  y OR 6  p OR 6  w 8! (5)—— = 4! 4! FINISH THIS EXERCISE. THIS HANDOUT IS PART OF THE NEXT HOMEWORK. #U = #E = 6 5 = 7776

12. 6/4/2016MATH 106, Section 1912 U = E = {all 5-tuples of the integers 1 to 6, repetition possible} {all integer solutions to r + b + y + p + w = 15 with 1  r  6 AND 1  b  6 AND 1  y  6 AND 1  p  6 AND 1  w  6} r + b + y + p + w = 10 r  5 AND b  5 AND y  5 AND p  5 AND w  5 Use GOOD = ALL – BAD r + b + y + p + w = 10 6  r OR 6  b OR 6  y OR 6  p OR 6  w 8! (5)—— = 350 4! 4! 14! ——– – 350 = 651 4! 10! #U = #E = 6 5 = 7776 651 #E P(E) = —– = #U 651 —— = 0.084 or 8.4% 7776

13. 6/4/2016MATH 106, Section 1913 Homework Hints: In Section 20 Homework Problems #3 and #4, the approach that can be used is any of the approaches used in parts (e) and (f) of #1 on the Section #20 Handout. Quiz #6 COMING UP! Be sure to do the review problems for this, quiz posted on the internet. The link can be found in the course schedule.

14. 6/4/2016MATH 106, Section 1914 Final Exam - NEXT WEEK! The final exam is worth 400 points, twice as much as each of the semester exams. The remaining homework assignments count toward 50 of these points. The 350 points on the exam administered next week cover the most recent material (Sections 19 and 20) and all previous material in the course. The colored review sheet that I distributed contains practice review exercises for the final exam. You may bring this colored review sheet to the final exam and use it during the exam. Write anything on it you want beforehand. There is a link on the webpage schedule to the numerical answers only for these practice review exercises. It is up to you to do the exercises and check that you can get the correct numerical answers. You can email me if you have any questions.

15. 6/4/2016MATH 106, Section 1915 Any questions about the homework due today? You may now check final answers to the Sample Final (page 253), and ask any questions about these. Note: all homework submitted this week and the last quiz will be available for students to pick from my secretary Sheran Swank before the final exam. Answers to Sample Final - page 253 1) (a) 720 (b) 120 (c) 48 (d) 480 (e) 0.333=33.3% (f) 0.067=6.7% 2) (a) 5040 (b) 360 5) (a) 495 (b) 126 (c) 9 (d) 4096 (e) 377 7) (a) 10,626 (b) 3876 (c) 5985 (d) 9625 (e) 680 (f) 4984 9) (a) 28