Math Stats Probability
- 1. Table of Contents 1 I. Number & Quantity II. Algebra III. Functions IV. Geometry V. Statistics & Probability VI. Integrating Essential Skills
- 2. V. Statistics and Probability 2 A. Data Representation B. Statistical Interpretation C. Principles of Probability D. Frequency Tables and Venn Diagrams E. Other Probability Scenarios
- 3. 1. Bar Charts and Histograms Axes X-axis: variable Y-axis: frequency of its occurrence Question types range but often include questions about average, probability, or other statistical interpretation 3
- 4. 2. Pie Charts Each “slice” represents a datum (piece of data) and is proportionally-sized 4 Half of the respondents liked summer best
- 5. 2. Pie Charts – Central Angle Angle is proportional to the percentage of respondents in that particular category 5 There are 360 degrees in a circle 180° 90° ?° ?° In decimal form
- 7. #1 6 Answer: B Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 8. V. Statistics and Probability 7 A. Data Representation B. Statistical Interpretation C. Principles of Probability D. Frequency Tables and Venn Diagrams E. Other Probability Scenarios
- 9. Measures of Center 8 Example Find the mean, median, and mode of the following set: 2, 4, 5, 4, 9, 11, 7
- 10. Measures of Center Three traditional measures of “average” 8 Example Find the mean, median, and mode of the following set: 2, 4, 5, 4, 9, 11, 7
- 11. Measures of Center Three traditional measures of “average” Mean – what the ACT calls the “average” (sum divided by number) 8 Example Find the mean, median, and mode of the following set: 2, 4, 5, 4, 9, 11, 7
- 12. Measures of Center Three traditional measures of “average” Mean – what the ACT calls the “average” (sum divided by number) Median – the middle of an ordered list of data points Think of the median in the road – in the middle 8 Example Find the mean, median, and mode of the following set: 2, 4, 5, 4, 9, 11, 7
- 13. Measures of Center Three traditional measures of “average” Mean – what the ACT calls the “average” (sum divided by number) Median – the middle of an ordered list of data points Think of the median in the road – in the middle Mode – the most frequently- occurring piece of data 8 Example Find the mean, median, and mode of the following set: 2, 4, 5, 4, 9, 11, 7
- 15. Other Statistical Values Range The area of variation between upper and lower limits on a set of numbers 9
- 16. Other Statistical Values Range The area of variation between upper and lower limits on a set of numbers Outlier A point that falls more than 1.5 times the interquartile range above the third quartile or below the first quartile 9
- 17. Other Statistical Values Range The area of variation between upper and lower limits on a set of numbers Outlier A point that falls more than 1.5 times the interquartile range above the third quartile or below the first quartile Do outliers affect the mean or the median more? 9
- 18. Other Statistical Values Range The area of variation between upper and lower limits on a set of numbers Outlier A point that falls more than 1.5 times the interquartile range above the third quartile or below the first quartile Do outliers affect the mean or the median more? Standard Deviation The measure of dispersion of a set of data from its mean When data are more spread out (further from the mean), the standard deviation increases 9
- 19. Other Statistical Values Range The area of variation between upper and lower limits on a set of numbers Outlier A point that falls more than 1.5 times the interquartile range above the third quartile or below the first quartile Do outliers affect the mean or the median more? Standard Deviation The measure of dispersion of a set of data from its mean When data are more spread out (further from the mean), the standard deviation increases 9
- 21. #2 10 Answer: B Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 22. V. Statistics and Probability 11 A. Data Representation B. Statistical Interpretation C. Principles of Probability D. Frequency Tables and Venn Diagrams E. Other Probability Scenarios
- 23. 1. Counting the Possibilities 12
- 24. 1. Counting the Possibilities Fundamental Counting Principle If there are m ways one “event” can happen and n ways a second “event” can happen, then there are m x n ways for both events to happen. 12
- 25. 1. Counting the Possibilities Fundamental Counting Principle If there are m ways one “event” can happen and n ways a second “event” can happen, then there are m x n ways for both events to happen. Why? Think about this example. I have two shirts and two pairs of shorts. How many possible combinations do I have? 12
- 26. 1. Counting the Possibilities Fundamental Counting Principle If there are m ways one “event” can happen and n ways a second “event” can happen, then there are m x n ways for both events to happen. Why? Think about this example. I have two shirts and two pairs of shorts. How many possible combinations do I have? 121 2 3 4
- 27. 1. Counting the Possibilities Fundamental Counting Principle If there are m ways one “event” can happen and n ways a second “event” can happen, then there are m x n ways for both events to happen. Why? Think about this example. I have two shirts and two pairs of shorts. How many possible combinations do I have? 121 2 3 4 OR: 2 x 2 = 4
- 29. #3 13 Answer: D Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 30. 2. Probability (Single Event) 14
- 31. 2. Probability (Single Event) Probability is the chance of the occurrence of a certain event and is expressed in a fraction, decimal, or percentage Fraction: between 0 and 1 Decimal: between 0 and 1 Percentage: 0% and 100% 14
- 32. 2. Probability (Single Event) Probability is the chance of the occurrence of a certain event and is expressed in a fraction, decimal, or percentage Fraction: between 0 and 1 Decimal: between 0 and 1 Percentage: 0% and 100% 14
- 33. 2. Probability (Single Event) Probability is the chance of the occurrence of a certain event and is expressed in a fraction, decimal, or percentage Fraction: between 0 and 1 Decimal: between 0 and 1 Percentage: 0% and 100% 14 Favorable refers to the event for which you are finding the probability
- 34. 2. Probability (Single Event) Probability is the chance of the occurrence of a certain event and is expressed in a fraction, decimal, or percentage Fraction: between 0 and 1 Decimal: between 0 and 1 Percentage: 0% and 100% 14 May require the Fundamental Counting Principle Favorable refers to the event for which you are finding the probability
- 35. Probability (Single Event) Example A bag contains 5 blue marbles, 7 yellow marbles, 3 orange marbles, and 6 green marbles. If a blue marble is selected on the first pick, what is the probability that another blue marble will be selected on the second pick? 15
- 37. #4 16 Answer: C Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 38. 3. Probability (Multiple Events) 17
- 39. 3. Probability (Multiple Events) This is the probability that two independent (non-related) events occur – found by finding the product of individual probabilities for each event 17
- 40. 3. Probability (Multiple Events) This is the probability that two independent (non-related) events occur – found by finding the product of individual probabilities for each event Example Find the probability of rolling two standard dice and getting a six on each die. 17
- 41. 3. Probability (Multiple Events) This is the probability that two independent (non-related) events occur – found by finding the product of individual probabilities for each event Example Find the probability of rolling two standard dice and getting a six on each die. 17 Probability on Die #1: Probability on Die #2: Probability of both events:
- 43. #5 18 Answer: E Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 44. V. Statistics and Probability 19 A. Data Representation B. Statistical Interpretation C. Principles of Probability D. Frequency Tables and Venn Diagrams E. Other Probability Scenarios
- 46. Frequency Tables Frequency Table Constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies 20
- 47. Frequency Tables Frequency Table Constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies 20
- 48. Frequency Tables Frequency Table Constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies Two-Way Frequency Table Shows the observed number or frequency for two variables Rows indicate one category and columns indicate another 20
- 49. Frequency Tables Frequency Table Constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies Two-Way Frequency Table Shows the observed number or frequency for two variables Rows indicate one category and columns indicate another 20
- 51. #6 21 Answer: C Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 53. #7 22 Answer: A Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 54. Venn Diagrams 23
- 55. Venn Diagrams A diagram representing mathematical sets pictorially 23
- 56. Venn Diagrams A diagram representing mathematical sets pictorially Distinct sets can be found in separate circles 23
- 57. Venn Diagrams A diagram representing mathematical sets pictorially Distinct sets can be found in separate circles Common elements in the sets are represented by overlapping areas 23
- 58. Venn Diagrams A diagram representing mathematical sets pictorially Distinct sets can be found in separate circles Common elements in the sets are represented by overlapping areas Could see a two-circle or three- circle Venn Diagram on the ACT! 23
- 60. #8 24 Answer: C Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 61. V. Statistics and Probability 25 A. Data Representation B. Statistical Interpretation C. Principles of Probability D. Frequency Tables and Venn Diagrams E. Other Probability Scenarios
- 62. Permutations Definition: one of several possible variations in which a set or number of things can be arranged or ordered 26
- 63. Permutations Definition: one of several possible variations in which a set or number of things can be arranged or ordered 26
- 64. Permutations Definition: one of several possible variations in which a set or number of things can be arranged or ordered 26
- 66. #9 27 Answer: D Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
- 67. Expected Value Definition: a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence 28
- 68. Expected Value Definition: a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence 28
- 69. Expected Value Definition: a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence 28
- 71. #10 29 Answer: C Calculator? Answer Choice Approach? Drawing? Hypothetical Numbers?
Editor's Notes
- #2: Teacher Notes The Statistics and Probability section focuses on basic stats and probability concepts, including visual representations and simple computations. **We recommend that students take notes in the form of an outline, beginning with Roman numeral five(V) for Statistics and Probability and on the next slide, letter A**
- #4: Teacher Notes What’s the difference between a bar chart and a histogram? It’s all in the x-axis. A bar chart describes a categorical (non-quantitative) variable like the example on the left and the example on page 741; the order of the categories can be changed, and the bars do not touch. The example on the right is a histogram because its x-axis describes a quantitative variable; the order of the “categories” cannot be changed, and the bars touch. For all intents and purposes, these can be treated similarly on the test. It is not important that students know this distinction; this explanation is provided in case students ask about the difference. http://www.css-resources.com/sample-bar-chart.jpg https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Black_cherry_tree_histogram.svg/220px-Black_cherry_tree_histogram.svg.png
- #5: Teacher Notes The percentages in a pie chart should add up to 100%.
- #6: Teacher Notes 1. 10%: 36° and 15%: 54° (check to make sure that the sum of all central angles is 360°)
- #10: https://s-media-cache-ak0.pinimg.com/236x/4d/fe/ab/4dfeaba7c8de87fc761424d4537a8033.jpg Teacher Notes “Ordered” means that the numbers are organized from smallest to largest. How do you find the median if there is an even number of data points? Take the mean of the two numbers at the middle of the set of values. A set does not have to have a mode! (This occurs when there is no single value that occurs more frequently than all other values.) Additionally, the mean and median need not be values that are actually part of the data set. Answers: Mean – 6 Median – 5 Mode – 4
- #11: https://s-media-cache-ak0.pinimg.com/236x/4d/fe/ab/4dfeaba7c8de87fc761424d4537a8033.jpg Teacher Notes “Ordered” means that the numbers are organized from smallest to largest. How do you find the median if there is an even number of data points? Take the mean of the two numbers at the middle of the set of values. A set does not have to have a mode! (This occurs when there is no single value that occurs more frequently than all other values.) Additionally, the mean and median need not be values that are actually part of the data set. Answers: Mean – 6 Median – 5 Mode – 4
- #12: https://s-media-cache-ak0.pinimg.com/236x/4d/fe/ab/4dfeaba7c8de87fc761424d4537a8033.jpg Teacher Notes “Ordered” means that the numbers are organized from smallest to largest. How do you find the median if there is an even number of data points? Take the mean of the two numbers at the middle of the set of values. A set does not have to have a mode! (This occurs when there is no single value that occurs more frequently than all other values.) Additionally, the mean and median need not be values that are actually part of the data set. Answers: Mean – 6 Median – 5 Mode – 4
- #13: https://s-media-cache-ak0.pinimg.com/236x/4d/fe/ab/4dfeaba7c8de87fc761424d4537a8033.jpg Teacher Notes “Ordered” means that the numbers are organized from smallest to largest. How do you find the median if there is an even number of data points? Take the mean of the two numbers at the middle of the set of values. A set does not have to have a mode! (This occurs when there is no single value that occurs more frequently than all other values.) Additionally, the mean and median need not be values that are actually part of the data set. Answers: Mean – 6 Median – 5 Mode – 4
- #14: https://s-media-cache-ak0.pinimg.com/236x/4d/fe/ab/4dfeaba7c8de87fc761424d4537a8033.jpg Teacher Notes “Ordered” means that the numbers are organized from smallest to largest. How do you find the median if there is an even number of data points? Take the mean of the two numbers at the middle of the set of values. A set does not have to have a mode! (This occurs when there is no single value that occurs more frequently than all other values.) Additionally, the mean and median need not be values that are actually part of the data set. Answers: Mean – 6 Median – 5 Mode – 4
- #15: Teacher Notes Outliers affect the mean more.
- #16: Teacher Notes Outliers affect the mean more.
- #17: Teacher Notes Outliers affect the mean more.
- #18: Teacher Notes Outliers affect the mean more.
- #19: Teacher Notes Outliers affect the mean more.
- #20: Teacher Notes Outliers affect the mean more.
- #24: Teacher Notes “Event” is just a fancier way of indicating that something happens, whether it’s choosing a shirt or picking a route to the store. The Fundamental Counting Principle can also be extended to more than two events. Simply take the product of the number of ways each independent “event” can occur. The ACT frequently tests this concept.
- #25: Teacher Notes “Event” is just a fancier way of indicating that something happens, whether it’s choosing a shirt or picking a route to the store. The Fundamental Counting Principle can also be extended to more than two events. Simply take the product of the number of ways each independent “event” can occur. The ACT frequently tests this concept.
- #26: Teacher Notes “Event” is just a fancier way of indicating that something happens, whether it’s choosing a shirt or picking a route to the store. The Fundamental Counting Principle can also be extended to more than two events. Simply take the product of the number of ways each independent “event” can occur. The ACT frequently tests this concept.
- #27: Teacher Notes “Event” is just a fancier way of indicating that something happens, whether it’s choosing a shirt or picking a route to the store. The Fundamental Counting Principle can also be extended to more than two events. Simply take the product of the number of ways each independent “event” can occur. The ACT frequently tests this concept.
- #28: Teacher Notes “Event” is just a fancier way of indicating that something happens, whether it’s choosing a shirt or picking a route to the store. The Fundamental Counting Principle can also be extended to more than two events. Simply take the product of the number of ways each independent “event” can occur. The ACT frequently tests this concept.
- #31: Teacher Notes For the fractional representation of probabilities, note that the numerator can never be greater than the denominator.
- #32: Teacher Notes For the fractional representation of probabilities, note that the numerator can never be greater than the denominator.
- #33: Teacher Notes For the fractional representation of probabilities, note that the numerator can never be greater than the denominator.
- #34: Teacher Notes For the fractional representation of probabilities, note that the numerator can never be greater than the denominator.
- #35: Teacher Notes For the fractional representation of probabilities, note that the numerator can never be greater than the denominator.
- #36: http://museumvictoria.com.au/collections/itemimages/254/551/254551_large.jpg Teacher Notes Answer: 4/20 or 1/5 (must account for the marble removed on the first pick)
- #39: Teacher Notes The rolling of dice represents independent events because the first roll has no impact on the second roll.
- #40: Teacher Notes The rolling of dice represents independent events because the first roll has no impact on the second roll.
- #41: Teacher Notes The rolling of dice represents independent events because the first roll has no impact on the second roll.
- #42: Teacher Notes The rolling of dice represents independent events because the first roll has no impact on the second roll.
- #46: Teacher Notes For two-way frequency tables, note that the marginal frequencies must be the sum of the corresponding row/column.
- #47: Teacher Notes For two-way frequency tables, note that the marginal frequencies must be the sum of the corresponding row/column.
- #48: Teacher Notes For two-way frequency tables, note that the marginal frequencies must be the sum of the corresponding row/column.
- #49: Teacher Notes For two-way frequency tables, note that the marginal frequencies must be the sum of the corresponding row/column.
- #50: Teacher Notes For two-way frequency tables, note that the marginal frequencies must be the sum of the corresponding row/column.
- #63: SEE VIDEO LESSON FOR EXAMPLE
- #64: SEE VIDEO LESSON FOR EXAMPLE
- #65: SEE VIDEO LESSON FOR EXAMPLE
- #68: SEE VIDEO LESSON FOR EXAMPLE Teacher Notes The expected value of a roll based on the diagram from this slide is calculated as follows: (0)(1/6) + (0)(1/6) + (0)(1/6) + (0)(1/6) + (10)(1/6) + (20)(1/6) = 5 Each term represents the money that will be won if a particular number is rolled and the probability of rolling that number. We would expect, on average, to win $5 per roll.
- #69: SEE VIDEO LESSON FOR EXAMPLE Teacher Notes The expected value of a roll based on the diagram from this slide is calculated as follows: (0)(1/6) + (0)(1/6) + (0)(1/6) + (0)(1/6) + (10)(1/6) + (20)(1/6) = 5 Each term represents the money that will be won if a particular number is rolled and the probability of rolling that number. We would expect, on average, to win $5 per roll.
- #70: SEE VIDEO LESSON FOR EXAMPLE Teacher Notes The expected value of a roll based on the diagram from this slide is calculated as follows: (0)(1/6) + (0)(1/6) + (0)(1/6) + (0)(1/6) + (10)(1/6) + (20)(1/6) = 5 Each term represents the money that will be won if a particular number is rolled and the probability of rolling that number. We would expect, on average, to win $5 per roll.