Topic IV: Probability

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Topic IV Probability
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The Concept of Probability

• Probability is a measure of the likelihood or
  chance that an event will occur or that a
  statistical experiment will have a particular
  outcome
• There are three main approaches to probability
• a priori classical
• Empirical classical
• Subjective
• Probability is a fraction (proportion) whose
  values range between 0 (impossible event) and 1
  (certain event).

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The a priori Classical Approach

• This is where the probability of success is based
  on prior knowledge of the process involved
• Simplest case is where each outcome is equally
  likely
• For example, consider a standard pack of cards
  which has 26 red and 26 black cards. The
  probability of selecting a black card is 26/52
  0.5.

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The a priori Classical Approach

• This probability does not mean that if each card
  is replaced after it is drawn, then one out of
  the next two cards will be black. It means that,
  in the long run, if the selection process is
  continued, the proportion of black cards selected
  will approach 0.5.

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The Empirical Classical Approach

• In this approach, although probability is still
  defined as the ratio of the number of favourable
  outcomes to the total number of outcomes, the
  outcomes are based on observed data and not on
  prior knowledge of a process.

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The Empirical Classical Approach

• For example the number of individuals in a survey
  who prefer a particular political party or the
  number of students who bought lunch at KFC in the
  past week

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Subjective Probability Approach

• This is when probability is based on personal
  judgement or beliefs
• It may be quite different from the objective
  probability assigned to the event as well as it
  may be different from the subjective probability
  assigned by another person.

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Subjective Probability Approach

• It is usually based on a combination of past
  experiences, personal opinions and possible
  analysis of a particular situation
• For example the development team of a particular
  product may assign a probability of 0.6 to the
  chance of the product being successful, the
  president of the company on the other hand being
  more cautious may assign a probability of 0.3.

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Example 12

• A balanced coin was tossed 100 times and the
  results are
• Heads - 56 Tails - 44
• Calculate the Empirical Classical and the a
  priori Classical Probability in simplest form.

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Example 13

• The results of a balanced die being tossed 300
  times are as follows
• 1 51 4 51
• 2 54 5 49
• 3 48 6 47
• Calculate the Empirical Classical and the a
  priori Classical Probability in simplest form.

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Probability Concepts

• Events
• Simple Event a single outcome of a process, it
  can be described by a single characteristic
• Compound (Joint) Event a combination of two or
  more simple events
• A statistical experiment is a repeatable process
  which has two or more possible outcomes, with a
  probability assigned to each outcome

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Probability Concepts

• Sample Space the collection of all the possible
  outcomes of a statistical experiment
• Sample Point this is an individual outcome of a
  statistical experiment

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Probability Concepts

• Statistical Independence events are independent
  if the occurrence of one event does not affect
  the probability of occurrence of other event(s)
• Two events are said to be statistically
  independent if and only if
• 
  or if

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Probability Concepts

• Events are mutually exclusive events, if they
  cannot occur at the same time
• For mutually exclusive events their intersection
  is said to be the empty set

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Axioms of Probability

• If A is an event, then 0 P(A) 1. Probability
  is ALWAYS positive and cannot be greater than 1
• P(S) 1 - the sum of the probabilities of a set
  of mutually exclusive events adds to one
• The probability of the union of two simple events
  for mutually exclusive units is the sum of the
  individual probabilities

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Example 14

• A sample consists of five simple events E1, E2,
  E3, E4 and E5.
• If P(E1) P(E2) 0.15, P(E3) 0.4 P(E4)
  2P(E5) find the probability of E4 and E5
• If P(E1) 3P(E2) 0.3, find the probability of
  the remaining simple events if it is known that
  the remaining events are equally probable

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Probability Rules

• The complement of an event, A, includes all
  events that are not a part of event A
• If A and A are complementary events in sample
  space, S, then
• P(A) 1 P(A)
• P(?) 0 for any sample space S
• If A and B are events in a sample space, S, and A
  ? B then P(A) P(B)

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Probability Rules

• If A and B are events in sample space S, then

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Probability Rules

• Union (Or)
• a set containing all elements in either A, or B
  or both A and B
• Either
• Or
• At least
• At most

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Probability Rules

• Intersection (And)
• a set containing all elements in both A and B
• Joint occurrence
• Happening at the same time
• Simultaneously
• Both

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Probability Rules
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Probability Rules

• The Addition Rule of Probabilities
• Let A and B be two events. The probability of
  their union is
• Note that if the events are mutually exclusive
  then is the empty set and
  is zero. In this case, the probability of
  their union would be simply

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Probability Rules

• The Multiplication Rule of Probabilities
• Let A and B be two events. The probability of
  their intersection is given by
• If the two events are independent then

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Example 15

• If the probabilities are 0.87, 0.36 and 0.29
  that a family randomly chosen as part of a sample
  survey owns a colour TV set, or a video camera
  recorder, or both. What is the probability that
  a family in the area will own at least one of the
  two?

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Example 16

• A woman customer selects 2 hair dryers, one
  after the other, from a shelf containing 12 hair
  dryers, 3 of which are defective.
• What is the probability that both of them will be
  defective?
• What is the probability of the female customer
  selecting two non-defective hair dryers?
• (Note selection is without replacement)

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Types of Probabilities

• Marginal (simple) Probability this is the
  probability that one event will occur (Eg. the
  probability of event A)

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Types of Probabilities

• Joint Probability this is the probability that
  two or more events will occur at the same time
  (Eg the probability of A and B)
• Or

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Computing Joint and Marginal Probabilities

• The probability of a joint event, A and B
• Computing a marginal (or simple) probability
• Where B1, B2, , Bk are k mutually exclusive and
  collectively exhaustive events

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Joint Probability Example
P(Red and Ace)
Color
Type
Total
Black
Red
2
2
4
Ace
24
24
48
Non-Ace
26
26
52
Total
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Marginal Probability Example
P(Ace)
Color
Type
Total
Black
Red
2
2
4
Ace
24
24
48
Non-Ace
26
26
52
Total
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Joint Probabilities Using Contingency Table
Event
Total
B1
B2
Event

P(A1 and B2)
P(A1)
P(A1 and B1)
A1
P(A2 and B1)
A2
P(A2 and B2)
P(A2)
Total
1
P(B1)
P(B2)
Marginal (Simple) Probabilities
Joint Probabilities
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Computing Conditional Probabilities

• A conditional probability is the probability of
  one event, given that another event has occurred

The conditional probability of A given that B has
occurred
The conditional probability of B given that A has
occurred
Where P(A and B) joint probability of A and B
P(A) marginal probability of A P(B)
marginal probability of B
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Conditional Probability

• 
  
  

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Conditional Probability Example

• Of the cars on a used car lot, 70 have air
  conditioning (AC) and 40 have a CD player (CD).
  20 of the cars have both.

• What is the probability that a car has a CD
  player, given that it has AC ?
• i.e., we want to find P(CD AC)

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Conditional Probability Example
(continued)

• Of the cars on a used car lot, 70 have air
  conditioning (AC) and 40 have a CD player (CD).
  20 of the cars have both.

No CD
CD
Total
0.2
0.5
0.7
AC
0.2
0.1
No AC
0.3
0.4
0.6
1.0
Total
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Conditional Probability Example

• Given AC, we only consider the top row (70 of
  the cars). Of these, 20 have a CD player. 20
  of 70 is about 28.57.

No CD
CD
Total
0.2
0.5
0.7
AC
0.2
0.1
No AC
0.3
0.4
0.6
Total
1.0
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Example

• If a person is selected at random, what is the
  probability that (s)he gets a dealer who provides
  good service?
• If a person randomly selects a dealer, what is
  the probability that (s)he gets dealer who was
  in business for less than 10 years and also
  provides good service?
• If a person randomly selects a dealer who was in
  business for more than 10 years, what is the
  probability that (s)he gets one that provides
  good service?