Topic II: Introduction to Probability

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Topic II Introduction to Probability

• Basic Introduction to the Use of Statistical
  Analysis in Business

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Random Experiment

• A random experiment is a process which has two or
  more possible outcomes, with no certainty as to
  which outcome will occur.

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Examples of Random Experiments

• The toss of a coin, the outcome is either a head
  or a tail
• A customer enters a store and either makes a
  purchase or does not
• A die is rolled
• A box of cereal is selected from a production
  line and is weighed to see if the weight is
  correct.

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Sample Space

• Sample Space The possible outcomes of a random
  experiment are called the basic outcomes, and the
  set of all basic outcomes is called the sample
  space. The symbol S will be used to denote the
  sample space.
• Event An event, E, is any subset of basic
  outcomes from the Sample Space.

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Sample Space - Example

• What is the sample space for the roll of a single
  six-sided die?
• What is the sample space for the toss of a coin?

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Probability Rules for Basic Outcomes

• All basic outcome probabilities must lie between
  0 and 1.
• The probabilities of all the basic outcomes
  within a sample space must sum to 1.

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Steps for Calculating Probabilities of Events

• Define the experiment and list the basic outcomes
• Assign a probability to each basic outcome
• Determine the basic outcomes which are contained
  in the event of interest
• Add the basic outcome probabilities to get the
  event probability.

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Intersections and Unions of Events

• Let A and B be two events in the sample space S.
  The intersection of A and B , is
  the set of all basic outcomes in S that belong to
  both A and B.
• Let A and B be two events in the sample space S.
  The union of A and B , is the set
  of all basic outcomes in S that belong to either
  A or B.

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Mutually Exclusive Events

• If the events A and B have no common basic
  outcomes, they are mutually exclusive and their
  intersection A ? B is said to be the empty set .
• That is, if A and B are mutually exclusive
  events, then A and B cannot occur at the same
  time.

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Intersection of Events A and B
S
S
A
B
A
B
A?B
(a) A?B is the striped area
(b) A and B are Mutually Exclusive
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Union of Events A and B
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Collectively Exhaustive

• Given the K events E1, E2, . . ., EK in the
  sample space S. If E1 ? E2 ? . . . ?EK S, then
  these events are said to be collectively
  exhaustive.
• That is, if the union of several events covers
  the entire sample space, S, then the events are
  said to be collectively exhaustive.

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Complement

• Let A be an event in the sample space S. the set
  of basic outcomes of a random experiment which
  belong to S but do not belong to A is called A
  complement or the complement of A. A complement
  is denoted by A or .

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Venn Diagram for the Complement of Event A
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Example 1

• A die is rolled. Let A be the event Number
  rolled is even and B be the event Number rolled
  is at least 4. Then
• A 2, 4, 6 and B 4, 5, 6
• Find the complements of each event, the
  intersection and the union of A and B. Also find

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Definition of Probability

• Probability A numerical measure of the
  likelihood of an event occurring. The formula for
  finding the probability of an event is

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Combinations

• In some cases, counting all of the outcomes can
  be very time consuming if we had to identify all
  the outcomes. There is a formula for doing this.
• This formula calculates the number of
  combinations of n things taken k at a time.

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Probability Postulates

• Let S denote the sample space of a random
  experiment, Oi, the basic outcomes, and A, an
  event. For each event A of the sample space S,
  we assume that a number P(A) is defined and we
  have the postulates
• If A is any event in the sample space S, then
• Let A be an event in S, and let Oi denote the
  basic outcomes. Then
• 
  
  
  where the
  notation implies that the summation extends over
  all the basic outcomes in A.
• 3. P(S) 1

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Probability Rules

• Let A be an event and A its complement. The the
  complement rule is

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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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Venn Diagram for Addition Rule
P(A?B)
A
B

P(A)
P(B)
P(A?B)
A
B
A
B
A
B

-
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Probability Rules

• Conditional Probability
• Let A and B be two events. The conditional
  probability of event A, given that event B has
  occurred, is denoted by the symbol P(AB) and is
  found to be
• provided that P(B gt 0).

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Probability Rules

• Conditional Probability
• Let A and B be two events. The conditional
  probability of event B, given that event A has
  occurred, is denoted by the symbol P(BA) and is
  found to be
• provided that P(A gt 0).

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Example 2

• Let S be the sample space for the toss of a fair
  coin three times
• S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
• Let A be the event that heads comes up exactly
  once
• A HTT, THT, TTH
• Let B be the event that the first coin comes up
  heads
• B HHH, HHT, HTH, HTT.

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Example 2

• What is the probability that heads comes up
  exactly once, given that the first toss comes up
  heads?

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Probability Rules

• The Multiplication Rule of Probabilities
• Let A and B be two events. The probability of
  their intersection can be derived from the
  conditional probability as
• Also,

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Statistical Independence

• Two events are independent if the occurrence of
  one does not affect the occurrence of the other

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Statistical Independence

• Let A and B be two events. These events are said
  to be statistically independent if and only if
• From the multiplication rule it also follows that
• More generally, the events E1, E2, . . ., Ek are
  mutually statistically independent if and only if

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Example 3

• Suppose that 48 of all bachelor degrees in
  Jamaica are obtained by men and that 17.5 of all
  degrees are in business. Also 6 of all bachelor
  degrees go to men majoring in business. Are the
  events Bachelor degree holder is a man and
  Bachelor degree in business statistically
  independent?