1. Probability

2. The definition – probability of an Event
Applies only to the special case when
The sample space has a finite no.of outcomes, and
Each outcome is equi-probable
If this is not true a more general definition of probability is required.

3. Summary of the Rules of Probability

4. The additive rule and if P[A  B] = f P[A  B] = P[A] + P[B]
P[A  B] = P[A] + P[B] – P[A  B]
and
P[A  B] = P[A] + P[B]
if P[A  B] = f

5. The Rule for complements
for any event E

6. Conditional probability

7. The multiplicative rule of probability
and
if A and B are independent.
This is the definition of independent

8. Counting techniques

9. Summary of counting results
Rule 1
n(A1  A2  A3  …. ) = n(A1) + n(A2) + n(A3) + …
if the sets A1, A2, A3, … are pairwise mutually exclusive
(i.e. Ai  Aj = f)
Rule 2
N = n1 n2 = the number of ways that two operations can be performed in sequence if
n1 = the number of ways the first operation can be performed
n2 = the number of ways the second operation can be performed once the first operation has been completed.

10. Rule 3
N = n1n2 … nk
= the number of ways the k operations can be performed in sequence if
n1 = the number of ways the first operation can be performed
ni = the number of ways the ith operation can be performed once the first (i - 1) operations have been completed. i = 2, 3, … , k

11. Basic counting formulae
Orderings
Permutations
The number of ways that you can choose k objects from n in a specific order
Combinations
The number of ways that you can choose k objects from n (order of selection irrelevant)

12. Applications to some counting problems
The trick is to use the basic counting formulae together with the Rules
We will illustrate this with examples
Counting problems are not easy. The more practice better the techniques

13. Random Variables
Numerical Quantities whose values are determine by the outcome of a random experiment

14. Random variables are either
Discrete
Integer valued
The set of possible values for X are integers
Continuous
The set of possible values for X are all real numbers
Range over a continuum.

15. A die is rolled and X = number of spots showing on the upper face.
Examples
Discrete
A die is rolled and X = number of spots showing on the upper face.
Two dice are rolled and X = Total number of spots showing on the two upper faces.
A coin is tossed n = 100 times and X = number of times the coin toss resulted in a head.
We observe X, the number of hurricanes in the Carribean from April 1 to September 30 for a given year

16. Examples
Continuous
A person is selected at random from a population and X = weight of that individual.
A patient who has received who has revieved a kidney transplant is measured for his serum creatinine level, X, 7 days after transplant.
A sample of n = 100 individuals are selected at random from a population (i.e. all samples of n = 100 have the same probability of being selected) . X = the average weight of the 100 individuals.

17. The Probability distribution of A random variable
A Mathematical description of the possible values of the random variable together with the probabilities of those values

18. The probability distribution of a discrete random variable is describe by its :
probability function p(x).
p(x) = the probability that X takes on the value x.
This can be given in either a tabular form or in the form of an equation.
It can also be displayed in a graph.

19. A die is rolled and X = number of spots showing on the upper face.
Example 1
Discrete
A die is rolled and X = number of spots showing on the upper face. x 1 2 3 4 5 6
p(x)
1/6
formula
p(x) = 1/6 if x = 1, 2, 3, 4, 5, 6

20. Graphs
To plot a graph of p(x), draw bars of height p(x) above each value of x.
Rolling a die

21. Example 2
Two dice are rolled and X = Total number of spots showing on the two upper faces. x 2 3 4 5 6 7 8 9 10 11 12
p(x)
1/36
2/36
3/36
4/36
5/36
6/36
Formula:

22. Rolling two dice

23. 36 possible outcome for rolling two dice

24. Comments: Every probability function must satisfy:
1. The probability assigned to each value of the random variable must be between 0 and 1, inclusive:
2. The sum of the probabilities assigned to all the values of the random variable must equal 1: 3.

25. Example
In baseball the number of individuals, X, on base when a home run is hit ranges in value from 0 to 3. The probability distribution is known and is given below:
Note:
This chart implies the only values x takes on are 0, 1, 2, and 3.
If the random variable X is observed repeatedly the probabilities, p(x), represents the proportion times the value x appears in that sequence. P X ( )
the random
variable
equals 2 p = 2 3 14

26. A Bar Graph

27. Discrete Random Variables
Discrete Random Variable: A random variable usually assuming an integer value.
a discrete random variable assumes values that are isolated points along the real line. That is neighbouring values are not “possible values” for a discrete random variable
Note: Usually associated with counting
The number of times a head occurs in 10 tosses of a coin
The number of auto accidents occurring on a weekend
The size of a family

28. Continuous Random Variables
Continuous Random Variable: A quantitative random variable that can vary over a continuum
A continuous random variable can assume any value along a line interval, including every possible value between any two points on the line
Note: Usually associated with a measurement
Blood Pressure
Weight gain
Height

29. Random Variables
Numerical Quantities whose values are determine by the outcome of a random experiment

30. The probability distribution of a discrete random variable
The probability distribution of a discrete random variable is describe by its :
probability function p(x).
p(x) = the probability that X takes on the value x.
This can be given in either a tabular form or in the form of an equation.
It can also be displayed in a graph.

31. Example

32. Probability Distributions of Continuous Random Variables

33. Probability Density Function
The probability distribution of a continuous random variable is describe by probability density curve f(x).

34. The Total Area under the probability density curve is 1.
Notes:
The Total Area under the probability density curve is 1.
The Area under the probability density curve is from a to b is P[a < X < b].

35. Normal Probability Distributions (Bell shaped curve)

36. Mean and Variance (standard deviation) of a Discrete Probability Distribution
Describe the center and spread of a probability distribution
The mean (denoted by greek letter m (mu)), measures the centre of the distribution.
The variance (s2) and the standard deviation (s) measure the spread of the distribution.
s is the greek letter for s.

37. Mean, Variance (and standard deviation) of a Probability Distribution

38. Mean of a Discrete Random Variable
The mean, m, of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together:
Notes:
The mean is a weighted average of the values of X.
The mean is the long-run average value of the random variable.
The mean is centre of gravity of the probability distribution of the random variable

40. Variance and Standard Deviation
Variance of a Discrete Random Variable: Variance, s2, of a discrete random variable x is found by multiplying each possible value of the squared deviation from the mean, (x - m)2, by its own probability and then adding all the products together:
Standard Deviation of a Discrete Random Variable: The positive square root of the variance: s = 2

41. Example
The number of individuals, X, on base when a home run is hit ranges in value from 0 to 3.

42. Computing the mean: Note:
0.929 is the long-run average value of the random variable
0.929 is the centre of gravity value of the probability distribution of the random variable

43. Computing the variance:
Computing the standard deviation:

44. Random Variables
Numerical Quantities whose values are determine by the outcome of a random experiment

45. Random variables are either
Discrete
Integer valued
The set of possible values for X are integers
Continuous
The set of possible values for X are all real numbers
Range over a continuum.

46. The Probability distribution of A random variable
A Mathematical description of the possible values of the random variable together with the probabilities of those values

47. The probability distribution of a discrete random variable is describe by its :
probability function p(x).
p(x) = the probability that X takes on the value x.
This can be given in either a tabular form or in the form of an equation.
It can also be displayed in a graph.

48. Example
In baseball the number of individuals, X, on base when a home run is hit ranges in value from 0 to 3. The probability distribution is known and is given below:
Note:
This chart implies the only values x takes on are 0, 1, 2, and 3.
If the random variable X is observed repeatedly the probabilities, p(x), represents the proportion times the value x appears in that sequence. P X ( )
the random
variable
equals 2 p = 2 3 14

49. A Bar Graph

50. Probability Distributions of Continuous Random Variables

51. Probability Density Function
The probability distribution of a continuous random variable is describe by probability density curve f(x).

52. The Total Area under the probability density curve is 1.
Notes:
The Total Area under the probability density curve is 1.
The Area under the probability density curve is from a to b is P[a < X < b].

53. Mean, Variance and standard deviation of Random Variables
Numerical descriptors of the distribution of a Random Variable

54. Mean of a Discrete Random Variable
The mean, m, of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together:
Notes:
The mean is a weighted average of the values of X.
The mean is the long-run average value of the random variable.
The mean is centre of gravity of the probability distribution of the random variable

55. Variance and Standard Deviation
Variance of a Discrete Random Variable: Variance, s2, of a discrete random variable x is found by multiplying each possible value of the squared deviation from the mean, (x - m)2, by its own probability and then adding all the products together:
Standard Deviation of a Discrete Random Variable: The positive square root of the variance: s = 2

56. Example
The number of individuals, X, on base when a home run is hit ranges in value from 0 to 3.

57. Computing the mean: Note:
0.929 is the long-run average value of the random variable
0.929 is the centre of gravity value of the probability distribution of the random variable

58. Computing the variance:
Computing the standard deviation:

59. The Binomial distribution
An important discrete distribution

60. Situation - in which the binomial distribution arises
We have a random experiment that has two outcomes
Success (S) and failure (F)
p = P[S], q = 1 - p = P[F],
The random experiment is repeated n times independently
X = the number of times S occurs in the n repititions
Then X has a binomial distribution

61. Example
A coin is tosses n = 20 times
X = the number of heads
Success (S) = {head}, failure (F) = {tail
p = P[S] = 0.50, q = 1 - p = P[F]= 0.50
An eye operation has %85 chance of success. It is performed n =100 times
X = the number of Sucesses (S)
p = P[S] = 0.85, q = 1 - p = P[F]= 0.15
In a large population %30 support the death penalty. A sample n =50 indiviuals are selected at random
X = the number who support the death penalty (S)
p = P[S] = 0.30, q = 1 - p = P[F]= 0.70

62. The Binomial distribution
We have an experiment with two outcomes – Success(S) and Failure(F).
Let p denote the probability of S (Success).
In this case q=1-p denotes the probability of Failure(F).
This experiment is repeated n times independently.
X denote the number of successes occuring in the n repititions.

63. The possible values of X are
0, 1, 2, 3, 4, … , (n – 2), (n – 1), n
and p(x) for any of the above values of x is given by:
X is said to have the Binomial distribution with parameters n and p.

64. Summary:
X is said to have the Binomial distribution with parameters n and p.
X is the number of successes occurring in the n repetitions of a Success-Failure Experiment.
The probability of success is p.
The probability function

65. Example:
A coin is tossed n = 5 times. X is the number of heads occurring in the 5 tosses of the coin. In this case p = ½ and x 1 2 3 4 5
p(x)

66. Note:

68. Computing the summary parameters for the distribution – m, s2, s

69. Computing the mean:
Computing the variance:
Computing the standard deviation:

70. Example:
A surgeon performs a difficult operation n = 10 times.
X is the number of times that the operation is a success.
The success rate for the operation is 80%. In this case p = 0.80 and
X has a Binomial distribution with n = 10 and p = 0.80.

71. Computing p(x) for x = 0, 1, 2, 3, … , 10

72. The Graph

73. Computing the summary parameters for the distribution – m, s2, s

74. Computing the mean:
Computing the variance:
Computing the standard deviation:

75. Notes
The value of many binomial probabilities are found in Tables posted on the Stats 245 site.
The value that is tabulated for n = 1, 2, 3, …,20; 25 and various values of p is:
Hence
The other table, tabulates p(x). Thus when using this table you will have to sum up the values

76. Example
Suppose n = 8 and p = 0.70 and we want to compute P[X = 5] = p(5)
Table value for n = 8, p = 0.70 and c =5 is = P[X ≤ 5]
P[X = 5] = p(5) = P[X ≤ 5] - P[X ≤ 4] = – = .254

77. We can also compute Binomial probabilities using Excel
The function
=BINOMDIST(x, n, p, FALSE)
will compute p(x).
The function
=BINOMDIST(c, n, p, TRUE)
will compute

78. Mean, Variance and standard deviation of Binomial Random Variables

79. Mean, Variance and standard deviation of Discrete Random Variables

80. Mean of a Discrete Random Variable
The mean, m, of a discrete random variable x
Notes:
The mean is a weighted average of the values of X.
The mean is the long-run average value of the random variable.
The mean is centre of gravity of the probability distribution of the random variable

81. Variance and Standard Deviation
Variance of a Discrete Random Variable: Variance, s2, of a discrete random variable x
Standard Deviation of a Discrete Random Variable: The positive square root of the variance: s = 2

82. The Binomial distribution

83. The Binomial distribution
X is said to have the Binomial distribution with parameters n and p.
X is the number of successes occurring in the n repetitions of a Success-Failure Experiment.
The probability of success is p.
The probability function

84. Mean,Variance & Standard Deviation of the Binomial Ditribution
The mean, variance and standard deviation of the binomial distribution can be found by using the following three formulas:

85. Example:
Find the mean and standard deviation of the binomial distribution when n = 20 and p = 0.75
Solutions:
1) n = 20, p = 0.75, q = = 0.25 m = np (
)(0. ) 20 75 15 s
npq . 25 3 1
936 p x ( )
(0. = æ è ç ö ø ÷ - 20 75 25
for 0, 1, 2, . ,
2) These values can also be calculated using the probability function:

86. Table of probabilities

87. Computing the mean:
Computing the variance:
Computing the standard deviation:

88. Histogram m s

89. Probability Distributions of Continuous Random Variables

90. Probability Density Function
The probability distribution of a continuous random variable is describe by probability density curve f(x).

91. The Total Area under the probability density curve is 1.
Notes:
The Total Area under the probability density curve is 1.
The Area under the probability density curve is from a to b is P[a < X < b].

92. Mean of a Continuous Random Variable (uses calculus)
The mean, m, of a discrete random variable x
Notes:
The mean is a weighted average of the values of X.
The mean is the long-run average value of the random variable.
The mean is centre of gravity of the probability distribution of the random variable

93. Variance and Standard Deviation
Variance of a Continuous Random Variable: Variance, s2, of a discrete random variable x
Standard Deviation of a Discrete Random Variable: The positive square root of the variance: s = 2

94. Normal Probability Distributions

95. Normal Probability Distributions
The normal probability distribution is the most important distribution in all of statistics
Many continuous random variables have normal or approximately normal distributions

96. The Normal Probability Distribution
Points of Inflection

97. Main characteristics of the Normal Distribution
Bell Shaped, symmetric
Points of inflection on the bell shaped curve are at m – s and m + s. That is one standard deviation from the mean
Area under the bell shaped curve between m – s and m + s is approximately 2/3.
Area under the bell shaped curve between m – 2s and m + 2s is approximately 95%.

98. There are many Normal distributions
depending on by m and s
Normal m = 100, s =20
Normal m = 100, s = 40
Normal m = 140, s =20

99. The Standard Normal Distribution m = 0, s = 1

100. There are infinitely many normal probability distributions (differing in m and s)
Area under the Normal distribution with mean m and standard deviation s can be converted to area under the standard normal distribution
If X has a Normal distribution with mean m and standard deviation s than
has a standard normal distribution.
z is called the standard score (z-score) of X.

101. under the Normal distribution with mean m and standard deviation s
Converting Area
under the Normal distribution with mean m and standard deviation s to
Area under the standard normal distribution

102. Perform the z-transformation
Area under the Normal distribution with mean m and standard deviation s
then
Area under the standard normal distribution

103. Area under the Normal distribution with mean m and standard deviation s

104. Area under the standard normal distribution1

105. Using the tables for the Standard Normal distribution

106. Table, Posted on stats 245 web site
The table contains the area under the standard normal curve between -∞ and a specific value of z

107. Example
Find the area under the standard normal curve between z = -∞ and z = 1.45
A portion of Table 3: z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
1.4
0.9265 .

108. Example Find the area to the left of -0.98; P(z < -0.98) P z ( 0. )= 98 .
1635

109. Example
Find the area under the normal curve to the right of z = 1.45; P(z > 1.45)

110. Example
Find the area to the between z = 0 and of z = 1.45; P(0 < z < 1.45)
Area between two points = differences in two tabled areas

111. Notes
Use the fact that the area above zero and the area below zero is
the area above zero is
When finding normal distribution probabilities, a sketch is always helpful

112. Example:
Find the area between the mean (z = 0) and z = -1.26

113. Example: Find the area between z = -2.30 and z = 1.80

114. Example: Find the area between z = -1.40 and z = -0.50

115. Convert the random variable, X, to its z-score.
Computing Areas under the general Normal Distributions (mean m, standard deviation s)
Approach:
Convert the random variable, X, to its z-score.
Convert the limits on random variable, X, to their z-scores.
Convert area under the distribution of X to area under the standard normal distribution.

116. Example 1: Suppose a man aged 40-45 is selected at random from a population.
X is the Blood Pressure of the man.
X is random variable.
Assume that X has a Normal distribution with mean m =180 and a standard deviation s = 15.

117. The probability density of X is plotted in the graph below.
Suppose that we are interested in the probability that X between 170 and 210.

118. Let
Hence

121. Example 2
A bottling machine is adjusted to fill bottles with a mean of 32.0 oz of soda and standard deviation of Assume the amount of fill is normally distributed and a bottle is selected at random:
1) Find the probability the bottle contains between oz and oz
2) Find the probability the bottle contains more than oz

122. Solution part 1)
When x = 32.00
When x =

123. Graphical Illustration:X z ( . ) 0.
32.0 32
025 02 1 25
3944
< = - æ è ç ö ø ÷

124. Example 2, Part 2) P x z ( . ) > = - æ è ç ö ø ÷ 31 97 32.0 0. 02 1
50)
0000
0668
9332

125. Summary Random Variables
Numerical Quantities whose values are determine by the outcome of a random experiment

126. Types of Random Variables
Discrete
Possible values integers
Continuous
Possible values vary over a continuum

127. The Probability distribution of a random variable
A Mathematical description of the possible values of the random variable together with the probabilities of those values

128. The probability distribution of a discrete random variable is describe by its :
probability function p(x).
p(x) = the probability that X takes on the value x.

130. The Binomial distribution
X is said to have the Binomial distribution with parameters n and p.
X is the number of successes occurring in the n repetitions of a Success-Failure Experiment.
The probability of success is p.
The probability function

131. Probability Distributions of Continuous Random Variables

132. Probability Density Function
The probability distribution of a continuous random variable is describe by probability density curve f(x).

133. The Total Area under the probability density curve is 1.
Notes:
The Total Area under the probability density curve is 1.
The Area under the probability density curve is from a to b is P[a < X < b].

134. The Normal Probability Distribution
Points of Inflection

135. The Standard Normal Distribution
m = 0, s = 1
Tables exist for the Standard Normal Distribution
These tables can be used for Normal distributions with different m and s .
The z transformation

136. Normal approximation to the Binomial distribution
Using the Normal distribution to calculate Binomial probabilities

137. Binomial distribution n = 20, p = 0.70
Approximating
Normal distribution
Binomial distribution

138. Normal Approximation to the Binomial distribution
X has a Binomial distribution with parameters n and p
Y has a Normal distribution

139. Approximating
Normal distribution
P[X = a]
Binomial distribution

141. P[X = a]

142. Example
X has a Binomial distribution with parameters n = 20 and p = 0.70

143. Where Y has a Normal distribution with:
Using the Normal approximation to the Binomial distribution
Where Y has a Normal distribution with:

144. Hence
= =
Compare with

145. Normal Approximation to the Binomial distribution
X has a Binomial distribution with parameters n and p
Y has a Normal distribution

148. Example
X has a Binomial distribution with parameters n = 20 and p = 0.70

149. Where Y has a Normal distribution with:
Using the Normal approximation to the Binomial distribution
Where Y has a Normal distribution with:

150. Hence
= =
Compare with

151. Comment:
The accuracy of the normal appoximation to the binomial increases with increasing values of n

152. Normal Approximation to the Binomial distribution
X has a Binomial distribution with parameters n and p
Y has a Normal distribution

153. Example
The success rate for an Eye operation is 85%
The operation is performed n = 2000 times
Find
The number of successful operations is between 1650 and 1750.
The number of successful operations is at most 1800.

154. Solution where Y has a Normal distribution with:
X has a Binomial distribution with parameters n = 2000 and p = 0.85
where Y has a Normal distribution with:

155. = =

156. Solution – part 2.
= 1.000

157. Next topic: Sampling Theory