The Basic Probability Model

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The Basic Probability Model

• The probability of an event is the relative
  likelihood of occurrence of an event or outcome.
  
• The probability of an event A is written P(A)
• The relative frequency is the frequency of some
  event occurring relative to all possible relevant
  events. Relative frequency is defined as the
  occurrence of some event relative to all the
  things that could have happened under similar
  circumstances.
• P(A) frequency of event A/ total of
  events/outcomes

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Basic Rules of Probability

• Rule 1 0 lt P(A) lt 1
• Rule 2 P S 1
• Rule 3 Simple additive rule for disjoint
  (mutually exclusive) events
• P(A or B) P(A) P(B)
• Rule 4 The complement rule
• P(Ac) 1 P(A)
• Rule 5 Simple multiplicative rule for
  independent events in sequence.
• P(A and B) P(A) P(B)

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Essence of all Probability Models

• Need a description of all possible outcomes and
• Must assign probabilities to each of the
  possible outcomes.

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Rule 1. The probability of any event P(A), is a
number between 0 and 1, meaning all events have
some probability of occurring. It may be small
or it may be large. Probabilities vary between
zero and one 0 lt P(A) lt 1 Rule 2. The
collection of all possible outcomes in a sample
space S has a probability of 1. The
probabilities of all elements in sample space
(all possible outcomes) sum to one PS 1.0
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Examples

• What is the probability of selecting the king of
  hearts from a complete deck of cards?
• P 1/52
• What is the probability of selecting an ace?
• P 4/52
• What is the probability of selecting a spade?
• P 13/52 ¼ or .25 or 25

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Rule 3. Simple additive rule for disjoint
(mutually exclusive) events P(A or B)
P(A) P(B) Rule 4. The Complement rule P(Ac)
1 P(A) Rule 5. Simple multiplicative rule for
independent events in sequence P(A and B) P(A)
P(B)
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Extension to Rule 3
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Illustration of Rule 3
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Illustration of Rule 4
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Illustration of Rule 5
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Additional Rules Rule 6. Additive rule for
events that are not mutually exclusive
events P(A or B) P(A) P(B) P(AB) Rule
7. Multiplicative rule for conditional
events P(A and B) P(A) p(BA)
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Illustration of Rule 6
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Rule 7
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Rule 7 (continued)
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Idealized distribution of grades in a class
The sample space all possibilities
SA,B,C,D,F P(S) P(A) P(B) P(C) P(D)
P(F) 1.0
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SONS CLASS OF A FATHER FROM LOWEST CLASS (SES)
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Probabilities Associated with Events that Occur
in Sequence

• There are two types
• Independent events
• And conditional sequences of events

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Independent Events

• Are events in a sequence but
• the occurrence of A does NOT effect the
  occurrence of B.
• interested in the occurrence of 2 or more events,
  each of which is independent (that is, not
  influencing) the occurrence of the other.

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Sampling Without Replacement

• Many events in the world are not independent.
• E.G., Independence is not the basic rule for card
  games where sequence matters.
• Knowing the outcome of the first event changes
  the probability of the second event.

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• What is the probability of drawing a flush, that
  is, 5 cards of the same suit?
• Where there are 4 suits, 13 cards in each.

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(13/52)(12/51)(11/50)(10/49)(9/48) .25
.235 .22 .204 .188
.0005 5 in 10 thousand chance of getting a
flush.
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Pick 6 numbers from a drum containing 40
consecutive numbers 1 thru 40. Whats the
probability of getting a 6 number jackpot? Are
these outcomes independent or conditional?
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P(event1) 6/40 .150 P(event2) 5/39
.128 P(event3) 4/38 .105 P(event4) 3/37
.081 P(event5) 2/36 .056 P(event6) 1/35
.029 P(all 6 events) P (event1 event2
event3 event4 event5 event6) (6/40)
(5/39) (4/48) (3/37) (2/36) (1/35)
.00000026
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• Ex) Consider a simple population containing only
  N10 scores with values 1,1,2,3,3,4,4,4,5,6.
• If you are taking a random sample of n1 score
  from this population, what is the probability of
  obtaining an individual with a score greater than
  4? What is the probability of selecting an
  individual with a score less than 5?

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To compute the mean of a random variable multiply
each event in the sample space by its probability
and then add up all the products. If X is
discrete random variable taking the values x1,
x2, , xk with probabilities of p1, p2, , pk,
the mean is computed as µx x1p1 x2p2
xkpk And the variance of x is (x1
µx)2p1 (x2 µx)2p2 (xk µx)2pk