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Algebra / Geometry II: Unit 4- Probability & Statistics SUCCESS CRITERIA: 1. Calculate theoretical probability involving simple events, length, and area 2. Calculate probability of independent events using the addition rule and Venn Diagrams. 3. Calculate probability of dependent events using the multiplication rule. 4. Evaluate outcomes using various probability concepts including bias. INSTRUCTOR: Craig Sherman Hidden Lake High School Westminster Public Schools PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org EMPOWER Recorded TARGET SCALE THEME MA.10.SP.01.04 Rules of Probability MA.10.SP.02.04 Independence & Conditional probability MA.10.SP.03.04 Evaluate Outcomes PROFICIENCY SCALE: SCORE 4.0 REQUIREMENTS In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class. Score 4.0 does not equate to more work but rather a higher level of performance. 3.5 In addition to Score 3.0 performance, in-depth inferences and applications with partial success. 3.0 The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught. o Calculate theoretical probability involving simple events, length, and area, AND o Calculate probability of independent events using the addition rule and Venn Diagrams, AND o Calculate probability of dependent events using the multiplication rule, AND o Evaluate outcomes using various probability concepts including bias. 2.0 Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as the learner… o Find probability of simple events with die and spinners, OR o Find probability of simple events with lengths, OR o Find probability of simple events with areas of figures, OR o Find probability using Venn Diagrams, OR o Use the Multiplication Rule to find the probability of dependent events, OR o Use two-way frequency tables to determine probability, OR o analyze decisions and strategies using probability. 1.0 Know and use the vocabulary Identify the Basic Elements With help, a partial understanding of some of the simpler details and process PMI-NJ Center for Teaching & Learning ~2~ NJCTL.org Simple Events: WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION outcomes simple events theoretical probability INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA Number of outcomes on a normal die = ______ Number of outcomes on a two die = ______ Number of outcomes on a four section spinner= ______ PMI-NJ Center for Teaching & Learning ~3~ NJCTL.org CLASSWORK You are rolling a standard number cube. Find the theoretical probability of each outcome. 1. P(rolling a 5) 2. P(rolling a 6) 3. P(rolling an even number) 4. P(rolling a 1) 5. P(rolling a multiple of 3) A bag contains 2 red ping-pong balls, 3 green ping-pong balls, 3 blue ping-pong balls, and 1 yellow pingpong ball. Find the probability of randomly selecting each outcome. 6. P(green) 7. P(not red) 8. P(red or blue) 9. P(red) Two standard number cubes are rolled. Find each probability. 10. P(a sum equal to 2) 11. P(sum not equal to 2) 12. P(a product less than or equal to 2) 13. P(a sum equal to 12) 14. A spinner has 4 equal sections. What is the theoretical probability of the spinner stopping on section 2? What is the sample space? 1 4 2 3 15. Is it possible for an event to have a probability of 1? Explain your answer. PMI-NJ Center for Teaching & Learning ~4~ NJCTL.org HOMEWORK You are rolling a standard number cube. Find the theoretical probability of each outcome. 16. P(rolling a 2) 17. P(rolling a 3) 18. P(rolling an odd number) 19. P(rolling a 7) 20. P(rolling a multiple of 2) A bag contains 1 red marble, 5 green marbles, 1 blue marble, and 1 yellow marble. Suppose one marble is picked at random. Find each probability. 21. P(blue) 22. P(not yellow) 23. P(red or green) 24. P(green) Two standard number cubes are rolled. Find each probability. 25. P(a sum equal to 3) 26. P(a sum not equal to 3) 27. P(a product equal to 15) 28. P(a sum greater than 6) 29. A game is played where students throw beanbags at the target shown to the right. Each region of the target is the same size and every beanbag hits the target. What is the theoretical probability of hitting section D? What is the sample space? A B C D 30. An event has a probability of 0. What does this tell you about the event? Explain. PMI-NJ Center for Teaching & Learning ~5~ NJCTL.org PROBABILITY and Length: CLASSWORK Find the probability that a point chosen at random from ̅̅̅̅ 𝐴𝐾 is on the given segment. ̅̅̅̅ 31. 𝐶𝐹 ̅̅̅ 32. 𝐵𝐼 33. ̅̅̅̅ 𝐺𝐾 ̅̅̅̅ 34. 𝐹𝐺 35. ̅̅̅̅ 𝐴𝐾 36. ̅̅̅̅ 𝐴𝐶 A point between A and B on each number line is chosen at random. What is the probability that the point is between C and D? 37. 38. 39. Roberto’s trolley runs every 45 minutes. If he arrives at the trolley stop at a random time, what is the probability that he will not have to wait more than 10 minutes? 40. Suppose that your school day begins at 7:30 am and ends at 3:00 pm. You eat lunch at 11:00 am. If there is a fire drill at a random time during the day, what is the probability that it is before lunch? PMI-NJ Center for Teaching & Learning ~6~ NJCTL.org HOMEWORK Find the probability that a point chosen at random from ̅̅̅̅ 𝐴𝐾 is on the given segment. ̅̅̅̅ 41. 𝐶𝐺 42. ̅̅̅̅ 𝐵𝐻 43. ̅̅̅̅ 𝐴𝐺 ̅̅̅̅ 44. 𝐴𝐵 45. ̅̅̅ 𝐴𝐽 46. ̅̅̅̅ 𝐷𝐹 A point between A and B on each number line is chosen at random. What is the probability that the point is between C and D? 47. 48. 49. A stoplight at an intersection stays red for 60 seconds, changes to green for 45 seconds, and then turns yellow for 15 seconds. If Jamal arrives at the intersection at a random time, what is the probability that he will have to wait at a red light? 50. You are expecting a call from a friend anytime between 6:00 pm and 7:00 pm. Unexpectedly you have to leave your house to run an errand from 5:45 to 6:10 pm. What is the probability that you missed your friend’s call? PMI-NJ Center for Teaching & Learning ~7~ NJCTL.org PROBABILITY and Area: AREA of a CIRCLE = ∏ r2 AREA of a RECTANGLE = l * w AREA of a TRIANGLE = 𝒃∗𝒉 AREA of a TRAPEZOID = 𝟐 (𝒃𝟏+𝒃𝟐) 𝒉 𝟐 CLASSWORK A point in the figure is chosen at random. Find the probability that the point lies in the shaded region. 51. 52. 53. Use the dartboard at the right for 54–56. Assume that a dart you throw will land on the dartboard and is equally likely to land at any point on the board. The radii of the concentric circles are 4 in, 6 in, and 8 in respectively. 54. What is the probability of hitting region X? 55. What is the probability of hitting region Y? 56. What is the probability of hitting region Z? PMI-NJ Center for Teaching & Learning ~8~ NJCTL.org SETS WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION Venn Diagram set { } union AUB intersection A∩B not ~A dependent mutually exclusive Addition Rule INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA INSTRUCTION3: KHAN ACADEMY Class Work PMI-NJ Center for Teaching & Learning ~9~ NJCTL.org 57. Use the Venn Diagram below to answer the questions. a. Name the Universe for the problem. b. Name all three of the sets involved. c. Find: i. A∩B ii. BUC iii. A∩(BUC) iv. ~(BUC)∩A v. ~(AUB) vi. A∩B∩C vii. AU(B∩C) viii. (AUB)∩~C 58. The following Venn Diagram shows the results of a college survey about extra-curricular activities that men (A) and women (B) do regularly. Use this diagram to list the members of the following sets and then write a sentence to describe the result. a. U b. A∩B c. A∩(~B) d. AUB e. ~A f. ~(A∩B) PMI-NJ Center for Teaching & Learning NJCTL.org ~10~ Home Work 59. Use the Venn Diagram below to answer the questions. a. Name the Universe for the problem. b. Name all three of the sets involved. c. Find: i. A∩B ii. BUC iii. A∩(AUC) iv. ~(A∩C)∩B v. ~(AUC) vi. A∩(BUC) vii. A∩B∩C viii. BU(A∩C) 60. The following Venn Diagram shows activities that male (A) and female (B) teens in Colorado do on the weekends. Use this diagram to list the members of the following sets and then write a sentence to describe the result. b. A∩B c. A∩(~B) d. AUB e. ~A f. ~(A∩B) PMI-NJ Center for Teaching & Learning NJCTL.org ~11~ Independence and Conditional PROBABILITY WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION independent probability conditional probability Multiplication Rule INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA INSTRUCTION 3: KHAN ACADEMY Multiplication Rule PMI-NJ Center for Teaching & Learning NJCTL.org ~12~ Class Work 61. Label the events as dependent or independent: a. Your family decides to take a trip to Disney World for spring break. Your friend’s family decides to go to Disneyland. b. You secretly take out all of the Aces from a deck of cards and then get your friend to see how many tries it takes to get an Ace. 62. Decide if the following events are mutually exclusive or overlapping. Then find P(AUB). a. A = Drawing a red card from a regular deck of cards B = Drawing a face card from a regular deck of cards b. A = Rolling an odd number on a six-sided die. B = Drawing a spade from a regular deck of cards 63. Find the conditional probability for the following problems: a. Find the probability that it is raining, given that it is cold. 64. Use the formula 𝑃(𝐵 |𝐴) 𝑃(𝐴) b. A bag contains different colored disks that are numbered from 1 to 10. The probability that the disk is green is 0.6. The probability that it is green and odd is 0.3. What is the probability that the disk is odd, given that it is green? = 𝑃(𝐵) to mathematically decide if the events are independent. a. Rolling a 10 on a set of six-sided die and then rolling a 5. b. A = taking math during your senior year at high school B = going to college PMI-NJ Center for Teaching & Learning NJCTL.org ~13~ Home Work 65. Label the events as dependent or independent: a. The cost of a person’s insurance is high. Looking at the person’s driving record, they have had a lot of accidents. b. You drink two 40oz. sodas a day for three weeks. In that time, you gain 15 pounds. 66. Decide if the following events are mutually exclusive or overlapping. Then find P(AUB). a. A bag of 15 marbles has 3 red marbles, 3 blue marbles, 3 yellow marbles, 3 green marbles and 3 black marbles. A = Drawing a red marble B = Drawing a blue marble b. Using a regular deck of cards: A = Drawing an even numbered card B = Drawing a heart 67. Find the conditional probability for the following problems: a. Find the probability that a student is in a band, given that they take a music class. 68. Use the formula 𝑃(𝐵 |𝐴) 𝑃(𝐴) b. Find the probability that a student gets good grades, given that they play a sport. = 𝑃(𝐵) to mathematically decide if the events are independent. a. The probability that a person owns the car they drive, i.e. no payments, in USA High School is 40%. The probability that the person owns their car and knows how to change the oil is 35%. Decide if the events are independent and then find the probability that a person at USA High knows how to change the oil on a car, given that they own the car. b. PMI-NJ Center for Teaching & Learning NJCTL.org ~14~ Two-Way Frequency Tables WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION relative frequency two-way frequency table INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA Class Work Put the following information in a two-way frequency table in the space below and use it to answer the questions. 69. At the humane society, during a summer, a total of 625 dogs and cats were brought in to see the veterinarian. Out of those animals, 145 dogs and 15 cats were positive for heartworm. 455 of the animals were dogs, the rest were cats. a. Find the probability that a dog came in tested positive for heartworm. b. What percentage of cats did not have heartworm? c. What is the probability that an animal who tested positive for heartworm was a dog? d. Find the probability that a cat came in and tested positive for heartworm. 70. At USA University, 300 students were interviewed about the sports that they played. Out of the 300, 170 were girls, the rest were boys. 121 of the girls and 103 of the boys played a sport. The rest did not play any sports. a. What percentage of girls played a sport? b. Find the probability that a person who played a sport was a boy. c. Of the boys, what percentage did not play a sport? PMI-NJ Center for Teaching & Learning NJCTL.org ~15~ d. Find the probability that a person who did not play a sport was a girl. Home Work Put the following information in a two-way frequency table in the space below and use it to answer the questions. 71. At USA College, 1000 students were surveyed about social websites. Half of the college students surveyed were girls. Out of the girls, 460 reported having an account on a social website. Of the boys, 320 stated that they had an account. a. What is the probability that a person who has a social account is a girl? b. Find the percentage of boys that do not have an account. c. Of the girls, what is the probability that they do not have a social account? d. Given that a person has a social account, what is the probability that they are a boy? 72. A group of students at a university studied leukemia in dogs and cats. 2000 pets participated in their study. 750 were dogs and the rest were cats. 57 of the dogs and 423 cats had leukemia. a. Did dogs have a greater percentage of leukemia or the cats? Prove your answer mathematically. b. Of the animals that had leukemia, what percentage were cats? c. Given that the animal was a dog, what is the probability that it would have leukemia? d. Find the probability that a dog would not have leukemia. PMI-NJ Center for Teaching & Learning NJCTL.org ~16~