1. Welcome to Week 06 College Statistics

2. Probability
Probability - likelihood of a favorable outcome

3. Probability is defined to be:
# favorable outcomes
total # of outcomes
Usually probabilities are given
in %
P =

4. Probability
This assumes each outcome is equally likely to occur – RANDOM

5. What is P(head on 1 toss of a fair coin)
Probability
IN-CLASS PROBLEMS
What is P(head on 1 toss of a fair coin)

6. P(head on 1 toss of a fair coin) What are all of the outcomes?
Probability
IN-CLASS PROBLEMS
P(head on 1 toss of a fair coin) What are all of the outcomes?

7. What are the favorable outcomes?
Probability
IN-CLASS PROBLEMS
P(head on 1 toss of a fair coin)
What are all of the outcomes?
H or T
What are the favorable outcomes?

8. Probability
IN-CLASS PROBLEMS
P(head on 1 toss of a fair coin) What are the favorable outcomes? H or T

9. So: P(head on 1 toss of a fair coin) = 1/2 or 50%
Probability
IN-CLASS PROBLEMS
So: P(head on 1 toss of a fair coin) = 1/2 or 50%

10. Probability The Law of Averages
And why it doesn’t work the way people think it does

11. Probability
IN-CLASS PROBLEMS
Amy Bob Carlos Dawn Ed ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE What is the probability that Bob will be going to the conference?

12. Probability
IN-CLASS PROBLEMS
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE How many outcomes total? How many outcomes favorable?

13. Probability
IN-CLASS PROBLEMS
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE How many outcomes total? 10 How many outcomes favorable? 6 have a “B” in them

14. Probability
IN-CLASS PROBLEMS
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE So there is a 6/10 or 60% probability Bob will be going to the conference

15. Probability
Because the # favorable outcomes is always less than the total # of outcomes
0 ≤ P ≤ 1 or
0% ≤ P ≤ 100%

16. Probability
And P(all possible outcomes) = 100%

17. Probability
The complement of an outcome is all outcomes that are not favorable Called P (pronounced “P prime") P = 1 – P or P = 100% – P

18. Probability
IN-CLASS PROBLEMS
To find: P (not 1 on one roll of a fair die) This is the complement of: P (a 1 on one roll of a fair die) = 1/6

19. Probability
IN-CLASS PROBLEMS
Since: P (a 1 on one roll of a fair die) = 1/6 Then P (not 1 on one roll of a fair die) would be: P = 1 – P = 1 – 1/6 = 5/6

20. Probability
Mutually exclusive outcomes – if one occurs, then the other cannot occur
Ex: if you have a “H” then you can’t have a “T”

21. Probability
Addition rule - if you have mutually exclusive outcomes: P(both) = P(first) + P(second)

22. Probability
Mutually exclusive events: Event A: I am 5’ 2” tall today Event B: I am 5’ 6” tall today

23. Probability The probability that a subscriber uses 1-20 min:
For tables of data, calculating probabilities is easy:
The probability that a subscriber uses 1-20 min:
P(1-20) = 0.17
Minutes Internet Usage
Probability of Being in Category
1-20
9/52 = 0.17 = 17%
21-40
18/52 = 0.35 = 35%
41-60
15/52 = 0.29 = 29%
61-80
0.15 = 15%
81-100
0.02 = 2%
0.00 = 0%
121+

24. Probability
IN-CLASS PROBLEMS
What is the probability of living more than 12 years after this diagnosis?
Years After
Diagnosis
% of Deaths
1-2 15
3-4 35
5-6 16
7-8 9
9-10 6
11-12 4
13-14 2
15+ 13

25. Probability
IN-CLASS PROBLEMS
Since you can’t live BOTH AND 15+ years after diagnosis (you fall into one category or the other) the events are mutually exclusive
Years After
Diagnosis
% of Deaths
1-2 15
3-4 35
5-6 16
7-8 9
9-10 6
11-12 4
13-14 2
15+ 13

26. Probability
IN-CLASS PROBLEMS
You can use the addition rule! P(living >12 years) = P(living13-14) + P(living 15+) = 2% + 13% = 15%
Years After
Diagnosis
% of Deaths
1-2 15
3-4 35
5-6 16
7-8 9
9-10 6
11-12 4
13-14 2
15+ 13

27. Probability
If two categories are “mutually exclusive” you add their probabilities to get the probability of one OR the other

28. Probability
Sequential outcomes – one after the other first toss: H second toss: T third toss: T . . .

29. Probability
Because one outcome in the sequence does not affect the outcome of the next event in the sequence we call them “independent outcomes”

30. Probability
Multiplication rule - if you have independent outcomes: P(one then another) = P(one) * P(another)

31. Probability
To get the probability of both event A AND event B occurring, you multiply their probabilities to get the probability of both

32. What if you had data: Natural Hair Color Eye Color Blonde 34.0% Blue
Probability
IN-CLASS PROBLEMS
What if you had data:
Natural Hair Color
Eye Color
Blonde
34.0%
Blue
36.0%
Brown
43.0%
64.0%
Red
7.0%
Black
14.0%

33. Probability
IN-CLASS PROBLEMS
If hair color and eye color are independent events, then find: P(blonde AND blue eyes)
Natural Hair Color
Eye Color
Blonde
34.0%
Blue
36.0%
Brown
43.0%
64.0%
Red
7.0%
Black
14.0%

34. P(blonde AND blue eyes) = =.34 x .36 ≈ .12 or 12%
Probability
IN-CLASS PROBLEMS
P(blonde AND blue eyes) = =.34 x .36 ≈ .12 or 12%
Natural Hair Color
Eye Color
Blonde
34.0%
Blue
36.0%
Brown
43.0%
64.0%
Red
7.0%
Black
14.0%

35. What would be the probability of a subscriber being both
(1-20 minutes)
and
(61-80 minutes)?
Minutes Internet Usage
Probability of Being in Category
1-20
9/52 = 0.17 = 17%
21-40
18/52 = 0.35 = 35%
41-60
15/52 = 0.29 = 29%
61-80
0.15 = 15%
81-100
0.02 = 2%
0.00 = 0%
121+

36. They are mutually exclusive categories
Probability
Zero!
They are mutually exclusive categories

37. Check out the gaming apparatus and answer the questions!
Probability
IN-CLASS PROBLEMS
Check out the gaming apparatus and answer the questions!

38. Probability Empirical vs theoretical probabilities
Empirical – based on observed
results
Theoretical – based on
mathematical theory

39. Probability
Are the probabilities of coin tosses empirical or theoretical?

40. Probability
Are the probabilities of horse racing empirical or theoretical?

41. Questions?

42. Probability
We were studying probability for things with outcomes like: H T Win Lose

43. Probability
These were all “discrete” outcomes

44. Probability
Now we will look at probabilities for things with an infinite number of outcomes that can be considered continuous

45. Probability
You can think of smooth quantitative data graphs as a series of skinnier and skinnier bars

46. Probability
When the width of the bars reach “zero” the graph is perfectly smooth

47. Probability
SO, a smooth quantitative (continuous) graph can be thought of as a bar chart where the bars have width zero

48. Probability
The probability for a continuous graph is the area of its bar: height x width

49. Probability
But… the width of the bars on a continuous graph are zero, so P = Bar Area = height x zero All the probabilities are P = 0 !

50. Probability
Yep. It’s true. The probability of any specific value on a continuous graph is: ZERO

51. Probability
So… Instead of a specific value, for continuous graphs we find the probability of a range of values – an area under the curve

52. Probability
Because this would require yucky calculus to find the probabilities, commonly-used continuous graphs are included in Excel Yay!

53. Normal Probability
The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

54. Normal Probability
Two descriptive statistics completely define the shape of a normal distribution: Mean µ Standard deviation σ

57. Suppose we have a normal distribution, µ = 12 σ = 2
Normal Probability
Suppose we have a normal distribution, µ = 12 σ = 2

58. Normal Probability
If µ = 12 12

59. Normal Probability
If µ = 12 σ = 2

60. ? Suppose we have a normal distribution, µ = 10 Normal Probability
PROJECT QUESTION
Suppose we have a normal distribution, µ = 10 ?

61. ? ? ? ? ? ? ? Suppose we have a normal distribution, µ = 10 σ = 5
Normal Probability
PROJECT QUESTION
Suppose we have a normal distribution, µ = 10 σ = 5
? ? ? ? ? ? ?

62. Normal Probability
Suppose we have a normal distribution, µ = 10 σ = 5

63. Questions?

64. Normal Probability
The standard normal distribution has a mean µ = 0 and a standard deviation σ = 1

65. ? ? ? ? ? ? ? For the standard normal distribution, µ = 0 σ = 1
Normal Probability
PROJECT QUESTION
For the standard normal distribution, µ = 0 σ = 1
? ? ? ? ? ? ?

66. -3 -2 -1 0 1 2 3 For the standard normal distribution, µ = 0 σ = 1
Normal Probability
PROJECT QUESTION
For the standard normal distribution, µ = 0 σ = 1

67. The standard normal is also called “z”
Normal Probability
The standard normal is also called “z”

68. Normal Probability
We can change any normally-distributed variable into a standard normal One with: mean = 0 standard deviation = 1

69. Normal Probability
To calculate a “z-score”: Take your value x Subtract the mean µ Divide by the standard deviation σ

70. Normal Probability
z = (x - µ)/σ

71. Normal Probability
IN-CLASS PROBLEMS
Suppose we have a normal distribution, µ = 10 σ = 2 z = (x - µ)/σ = (x-10)/2 Calculate the z values for x = 9, 10, 15

72. z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2
Normal Probability
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2

73. Normal Probability
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0

74. Normal Probability
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0 15 z = (15-10)/2 = 5/2

75. Normal Probability
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76. But… What about the probabilities??
Normal Probability
But… What about the probabilities??

77. Normal Probability
We use the properties of the normal distribution to calculate the probabilities

78. Normal Probability
IN-CLASS PROBLEMS
What is the probability of getting a z-score value between -1 and 1 -2 and 2 -3 and 3

79. You also use symmetry to calculate probabilities
Normal Probability
You also use symmetry to calculate probabilities

80. Normal Probability
IN-CLASS PROBLEMS
What is the probability of getting a z-score value between -1 & 0 -2 & 0 -3 & 0
Percentage
of the curve
z-score

81. Normal Probability
IN-CLASS PROBLEMS
What is the probability of getting a z-score value between -1 & & & -1.5
Percentage
of the curve
z-score

82. Normal Probability
Note: these are not exact – more accurate values can be found using Excel
Percentage
of the curve
z-score

83. Questions?

84. Excel Probability
Normal probability values in Excel are given as cumulative values (the probability of getting an x or z value less than or equal to the value you want)

85. Excel Probability
Use the function: NORM.DIST

86. Excel Probability

87. Excel Probability
For a z prob, use: NORM.S.DIST

88. Excel Probability
Excel gives you the cumulative probability – the probability of getting a value UP to your x value: P (x ≤ your value)

89. Excel Probability
How do you calculate other probabilities using the cumulative probability Excel gives you?

90. Normal Probability
IN-CLASS PROBLEMS
If the area under the entire curve is 100%, how much of the graph lies above “x”?

91. Excel Probability
P(x ≥ b) = 1 - P(x ≤ b) or = 100% - P(x ≤ b) or = 100% - 90% = 10%

92. What about probabilities between two “x” values?
Excel Probability
What about probabilities between two “x” values?

93. Excel Probability
P(a ≤ x ≤ b) equals P(x ≤ b) – P(x ≤ a) minus

94. Normal Probability
IN-CLASS PROBLEMS
Calculate the probabilities when µ = 0 σ = 1: P(-1 ≤ x ≤ 2) P(1 ≤ x ≤ 3) P(-2 ≤ x ≤ 1)

95. P(-1 ≤ x ≤ 2) = 34%+34%+13.5% = 81.5% Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 34%+34%+13.5% = 81.5%

96. P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 13.5%+2% = 15.5%
Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 13.5%+2% = 15.5%

97. Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 15.5% P(-3 ≤ x ≤ 1) = = 2%+13.5%+34%+34% = 83.5%

98. Questions?