Probability

1
Probability

• The Concept of Probability
• Sample Spaces and Events
• Some Elementary Probability Rules
• Conditional Probability and Independence

2
Probability

• An experiment is any process that generates
  well-defined outcomes.
• Experimental outcomes are the possible outcomes
  of an experiment.
• Probability is the numerical likelihood that an
  experimental outcome will occur.

3

• Probability values are always assigned on a scale
  from 0 to 1.
• A probability near 0 indicates an outcome is very
  unlikely to occur. If it equals 0 it will never
  occur.
• A probability near 1 indicates an outcome is
  almost certain to occur. If it equals 1 it will
  certainly occur.
• A probability of 0.5 indicates the occurrence of
  the outcome is just as likely as it is unlikely.
• The probabilities of all the experimental
  outcomes must sum to 1.

4
Methods of Assigning Probabilities

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

5
Classical Method

• If an experiment has n possible outcomes, this
  method
• would assign a probability of 1/n to each
  outcome.
• Example 4.1
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance
• of occurring.

6
Example 4.2 (Relative Frequency )

• Lucas would like to assign probabilities to the
  number of floor polishers rented per day. Office
  records show the following frequencies of daily
  rentals for the last 40 days.
• Number of Number
• Polishers Rented of Days
• 0 4
• 1 6
• 2 18
• 3 10
• 4 2

7
Example 4.2
The probability assignments are given by dividing
the number-of-days frequencies by the total
frequency
Number of Number Polishers
Rented of Days Probability 0 4
.10 4/40 1 6 .15 6/40 2
18 .45 etc. 3 10 .25 4
2 .05 40 1.00
Anderson, Sweeney, and Williams
8
Subjective Method

• When economic conditions and a companys
  circumstances change rapidly it might be
  inappropriate to assign probabilities based
  solely on historical data.
• We can use any data available as well as our
  experience and intuition, but ultimately a
  probability value should express our degree of
  belief that the experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

9
Sample Space and Events

• The sample space for an experiment is the set of
  all experimental outcomes.
• A sample point is an element of the sample space,
  any one particular experimental outcome.
• An event is a collection of sample points.
• The probability of any event is equal to the sum
  of the probabilities of the sample points in the
  event.
• If we can identify all the sample points of an
  experiment and assign a probability to each, we
  can compute the probability of an event.

10
Example 4.1 Revisited (Probabilities of Events)

• Recall the sample space for rolling a die is
  S1, 2, 3, 4, 5, 6
• Let AEvent that we roll a number less than 4
• A1,2,3 A is the event that we roll a 1, 2 ,
  or 3
• P(A) P(1) P(2) P(3) 1/6 1/6 1/6
• 3/6 1/2
• Let BEvent that we roll a number that is even
• B2,4,6
• P(B) P(2) P(4) P(6) 1/6 1/6 1/6 3/6
  1/2

11
Example 4.2 Revisited (Probabilities of Events)
Refer to slide 7. Let CEvent that Lucas rents
less than two polishers C0, 1 C is the event
that he rents 0 or 1 polishers P(C) P(0) P(1)
0.1 0.15 0.25
12
Event Relations

• Union of A and B
• The union of events A and B is the set of all
  sample points in the sample space that are in A
  or B or both.
• The union of events A and B is denoted A?B.
• Intersection of A and B
• The intersection of events A and B is the set of
  all sample points in the sample space that are in
  A and B.
• The intersection of events A and B is denoted
  A?B.
• Complement of E
• The complement of event E is the set of all
  sample points in the sample space that are not in
  E.
• The complement of E is denoted Ec .
• P(E) 1- P(Ec) (Law of Complements)

13
Example 4.1 Revisited (Intersection and Union)

• Compute P(A?B).
• A?B 1,2,3,4, 6 The sample points 1, 2, 3, 4
  and 6 are in A or B or both
• P(A?B) P(1) P(2) P(3) P(4) P(6)
• 1/6 1/6 1/6 1/6 1/6 5/6
• Compute P(A?B).
• A?B 2 The sample point 2 is in A and B
• P(A?B) P(2) 1/6
• Recall
• AEvent that we roll a number less than 4
• BEvent that we roll a number that is even

14
Complement

• Refer to slide 10 (Example 3.1). Compute P( Ac
  ).
• Ac 4, 5, 6
• P( Ac ) P(4) P(5) P(6) 1/6 1/6 1/6
  3/6 1/2
• According to the Law of Complements
• P(A) 1- P( Ac ) 1 1/2 1/2 As directly
  computed on slide 10.
• Refer to slide 11 (Example 3.2). Compute P(Cc ).
• Cc 2, 3, 4
• P(Cc ) P(2) P(3) P(4) 0.45 0.25 0.05
  0.75
• Thus,
• P(C) 1 0.75 0.25 As directly computed on
  slide 11
• Recall
• AEvent that we roll a number less than 4
• CEvent that Lucas rents less than two
  polishers

15
A Note About the Law of Complements

• The Law of Complements provide an alternate
  method for computing the probability of events.
  This technique is very useful when you want to
  compute the probability of an event and the
  complement of the event contains far fewer sample
  points than the actual events.

16
The Addition Rule for Unions

• The probability that A or B (the union of A and
  B) will occur is
• P(A?B) P(A) P(B) P(A?B)
• Two events are said to be mutually exclusive if
  the events have no sample points in common. That
  is, two events are mutually exclusive if, when
  one event occurs, the other cannot occur (i.e.
  P(A?B) 0)

17
Example 4.1 Revisited (Addition Rule for Unions)
Refer to slides 10 and 13. Using the additions
rule, P(A?B) P(A) P(B) - P(A?B) 1/2 1/2
- 1/6 5/6 As computed directly on slide
13. Either approach can be used to compute the
probability of the union of two events.
18
Conditional Probability
The probability of an event A, given that the
event B has occurred is called the conditional
probability of A given B and is denoted as
P(AB) . Moreover,
19
Example 4.1 Revisited (Conditional Probability)

• Refer to slides 10 and 13. Compute P(AB)
• P(AB)
• Explanation The P(Aevent that we roll a number
  less than five) equals 1/2. However, if we know
  that the number that we will roll will be an even
  number (i.e. we know that event B occurred), we
  must now determine P(AB) using this extra known
  information. Since there is only one sample
  point in B that is less than 4, there is a 1 in 3
  chance that the number that we roll will be less
  than 5.

20
Independence of Events

• Two events A and B are said to be independent if
  and only if
• P(AB) P(A) or, equivalently,
• P(BA) P(B)

21
The End