Probability

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• ??????? ????????????

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Definitions

• Random Experiment elementary outcomes
• Sample Space
• Event subset of sample space

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????? (Definition)

• An experiment is the process of observing a
  phenomenon that has variation in its outcomes.
• The sample space associated with an experiment is
  the collection of all possible distinct outcomes
  of the experiment.

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????? (Definition)

• An experiment is the process of observing a
  phenomenon that has variation in its outcomes.
• The sample space associated with an experiment is
  the collection of all possible distinct outcomes
  of the experiment.

SS
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????? (Definition) (cont.)

• Each outcome is called as elementary outcome, a
  simple event, or an elementary of the sample
  space.
• As event is the set of elementary outcomes
  possessing a designated feature.

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????? (Definition) (cont.)

• An event A occurs when any one of the elementary
  outcomes in A occurs.

(e1)
H
HH
H
(e2)
T
HT
(e3)
H
TH
T
(e4)
T
TT
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????? (Definition) (cont.)

• An event A occurs when any one of the elementary
  outcomes in A occurs.

(e1)
H
HH
H
(e2)
T
HT
(e3)
H
TH
T
(e4)
T
TT
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????? (Definition) (cont.)

• An event A occurs when any one of the elementary
  outcomes in A occurs.

(e1)
H
HH
H
(e2)
T
HT
(e3)
H
TH
T
(e4)
T
TT
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????? (Definition) (cont.)

• An event A occurs when any one of the elementary
  outcomes in A occurs.

(e1)
H
HH
H
(e2)
T
HT
(e3)
H
TH
T
(e4)
T
TT
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????? (Definition) (cont.)

• SS HH, HT, TH, TT
• SS e1, e2, e3, e4
• event A getting exactly one head
• A HH, HT
• A e1, e2

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????? (Definition) (cont.)

• Probability must satisfy

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Equally Likely Elementary Outcomes

• Uniform Probability Model
• SS e1, e2, e3, , ek

P(e1) P(e2) P(ek)
A e1, e2, e3, , em
P(A) P(e1) P(e2) P(em)
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Equally Likely Elementary Outcomes (cont.)

• Probability as Long-Run Relative Frequency

Define P(A) as the value to which the relative
frequency stabilizes with increasing of
trials. Although we will never know P(A) exactly,
it can be estimated accurately by repeating the
experiment many times.
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Equally Likely Elementary Outcomes (cont.)

• Tossing dice and coins
• drawing cards
• death in particular age groups
• etc.

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Equally Likely Elementary Outcomes (cont.)

• Consider of the probability to the day of the
  week a child will be born.
• Assume that all 7 days of week are equally
  likely.
• A event of a birth on the weekend
• P(A) 2/7 0.286

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Equally Likely Elementary Outcomes (cont.)

• Table Number of birth (in 10,000) by Day of the
  Week, U.S. 1971

Mon Tue Wed Thu Fri Sat Sun All Days
Number 52.09 54.46 52.68 51.68 53.83 47.21 44.36 3
56.31 of Births
Relative 0.146 0.153 0.148 0.145 0.151 0.132 0.124
Frequency
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Equally Likely Elementary Outcomes (cont.)

• Table Number of birth (in 10,000) by Day of the
  Week, U.S. 1971

0.132
Mon Tue Wed Thu Fri Sat Sun All Days
Number 52.09 54.46 52.68 51.68 53.83 47.21 44.36 3
56.31 of Births
Relative 0.146 0.153 0.148 0.145 0.151 0.132 0.124
Frequency
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Equally Likely Elementary Outcomes (cont.)

• Table Number of birth (in 10,000) by Day of the
  Week, U.S. 1971

0.132

0.124
Mon Tue Wed Thu Fri Sat Sun All Days
Number 52.09 54.46 52.68 51.68 53.83 47.21 44.36 3
56.31 of Births
Relative 0.146 0.153 0.148 0.145 0.151 0.132 0.124
Frequency
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Equally Likely Elementary Outcomes (cont.)

• Table Number of birth (in 10,000) by Day of the
  Week, U.S. 1971

0.256

0.132

0.124
Mon Tue Wed Thu Fri Sat Sun All Days
Number 52.09 54.46 52.68 51.68 53.83 47.21 44.36 3
56.31 of Births
Relative 0.146 0.153 0.148 0.145 0.151 0.132 0.124
Frequency
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Two Laws of Probability

• Law of Complement

• Addition Law

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Two Laws of Probability (cont.)

• Special Addition Law For Incompatible Events

P(AB) 0
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Two Laws of Probability (cont.)

• Special Addition Law For Incompatible Events

P(AB) 0
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Conditional Probability and Independence
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Conditional Probability and Independence (cont.)

• Multiplication Law of Probability

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Conditional Probability and Independence (cont.)

• Two events A and B are independent if

Equivalent conditions are
or
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Note
A and B are incompatible when

• A and B are independent if