Probability

1
Probability

• Part 2

2
Overview

• Counting Techniques in Probability
• Conditional Probability and Independent Events
• Bayes Theorem
• Distributions of Random Variables
• Expected Value, Variance and Standard Deviation

3
Probability of an Event in a Uniform Sample Space

• Let S be a uniform sample space (every sample
  point in S is equally likely to another) and let
  E be any event. Then,

4
Conditional Probability

• Consider the probability of an event, A, given
  that another event, B, has occurred. We call this
  the probability of A conditional on B, denoted
  P(A B).

5
Joint and Marginal Probabilities

• The joint probability of two events, is the
  probability of the intersection of these two
  events.
• Marginal probabilities provide the probabilities
  of each event separately.

6
Joint Probability Table
Payment Scheme
Joint Probability
On Meal Plan
Not On Meal Plan
Totals
Cash
0.79
0.39
0.40
Method of Payment
Debit
0.21
0.03
0.18
1.00
Totals
0.42
0.58
Marginal Probabilities
The above joint probability table provides data
on students using the school cafeteria at a
fictitious university.
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Interpreting the table

• There is a 58 probability that a student will
  not be on the meal plan
• There is a 79 probability that a student will
  use cash as method of payment
• The probability that a student will be on the
  meal plan and use cash as method of payment is
  0.39 or 39.
• The probability that a student will use cash
  given that the student is on the meal plans is
  0.39/0.420.93 or 93.

8
The Product Rule
Suppose that the probability that a professor is
in his/her office given that the door is ajar is
0.89. Suppose that the probability that the
professors office door is ajar is 0.32.
Therefore, the probability that a professor is
in his/her office and the door is ajar is equal
to (0.89)(0.32) 0.285
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Independent Events
Two events A and B are independent events if
Saying that two events are independent, does not
mean that the two events are mutually exclusive.
Two events A and B are dependent events if P(AB)
is not equal to P(A) and P(BA) is not equal to
P(B).
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Recall the school cafeteria example. Let A be the
event that a student is on the meal plan. Let B
be the event that a student uses cash as method
of payment.
Therefore, the events A and B are dependent.
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Independence of n Events
12
Prior and Posterior Probabilities

• Prior probabilities are the initial probabilities
  of events. (a.k.a. a priori probabilities)
• Posterior probabilities are revised probabilities
  based on additional information. (a.k.a. a
  posteriori probabilities)
• Posterior probabilities are found by calculating
  the probability after the outcome of an
  experiment has been observed.

13
Partitions of the Sample Space

• Sets A1, A2, An are partitions of the sample
  space, S, if they are mutually exclusive and
  their union is precisely S.

S
A
B
C
A, B and C are partitions of the sample space S.
AUBUC S
14
Bayes Theorem

• This is a method used to compute posterior
  probabilities.

Application of Bayes Theorem
Posterior Probabilities
Prior Probabilities
New Information
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S
B
C
The set D can be written as (DnA)U(DnB)U(DnC) So,
P(D) P(DnA)P(DnB)P(DnC)
A
D
16
Random Variables

• A random variable is a numerical description of
  the outcome of an experiment. Random variables
  must assume numerical values.
• A random variable is discrete if it may assume
  only a finite or an infinite sequence of
  values.(e.g. number of items sold)
• A random variable is continuous if it may assume
  any value within a certain interval or collection
  of intervals.(e.g. weight, temperature)

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Probability Distribution for Discrete Random
Variable

• The probability function, f(x), provides the
  probability that the random variable x takes on a
  specific value. (also denoted P(Xx))
• There are two requirements of a discrete
  probability distribution

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Histogram

• This is a graphical representation of the
  probability distribution of a discrete random
  variable. To create a histogram for a random
  variable,
• Locate the values of the random variables on the
  number line
• Then above each possible value, erect a rectangle
  with width 1 and height equal to the probability
  associated with that value of the random variable.

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• x P(Xx)
• 0 0.15
• 1 0.20
• 2 0.20
• 3 0.45

0.50
0.40
0.30
0.20
0.10
0
1
2
3
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Mean, Median and Mode
The median of a set of numbers arranged in
increasing or decreasing order is the middle
number if there is an odd number of entries, and
the mean of the two middle numbers if there is an
even number of entries. The mode of a set of
numbers is the number in the set that occurs most
frequently.
21
Expected Value

• The expected value of a discrete random variable
  is a weighted average of all possible values of
  the random variable.

22
Variance

• This is a measure of the dispersion or
  variability of the random variable, given the
  possible values it could take.

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Standard Deviation

• This is the square-root of the variance, allowing
  us to measure dispersion in the same units as the
  values of the random variable.

24
Chebychevs Inequality
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Probability Density for a Continuous Random
Variable

• This represents the height of the probability
  function at any particular value of the random
  variable and not the probability that the random
  variable takes that value.
• Since a continuous random variable has infinitely
  many values, we can no longer identify the
  probability of the random variable taking a
  specific value.
• Rather, we consider the probability the random
  variable takes on a value within a specified
  interval.