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

6. #1
6
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

7. #1
6
Answer: B
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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

14. Other Statistical Values
9

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

20. #2
10
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Answer Choice Approach?
Drawing?
Hypothetical Numbers?

21. #2
10
Answer: B
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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

28. #3
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29. #3
13
Answer: D
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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

36. #4
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37. #4
16
Answer: C
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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:

42. #5
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Answer Choice Approach?
Drawing?
Hypothetical Numbers?

43. #5
18
Answer: E
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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

45. Frequency Tables
20

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

50. #6
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Answer Choice Approach?
Drawing?
Hypothetical Numbers?

51. #6
21
Answer: C
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52. #7
22
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53. #7
22
Answer: A
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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

59. #8
24
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Answer Choice Approach?
Drawing?
Hypothetical Numbers?

60. #8
24
Answer: C
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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

65. #9
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66. #9
27
Answer: D
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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

70. #10
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71. #10
29
Answer: C
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