CHAPTERS 5

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CHAPTERS 5

• PROBABILITY, PROBABILITY RULES, AND CONDITIONAL
  PROBABILITY

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PROBABILITY MODELS FINITELY MANY OUTCOMES

• DEFINITION
• PROBABILITY IS THE STUDY OF RANDOM OR
  NONDETERMINISTIC EXPERIMENTS. IT MEASURES THE
  NATURE OF UNCERTAINTY.

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PROBABILISTIC TERMINOLOGIES

• RANDOM EXPERIMENT
• AN EXPERIMENT IN WHICH ALL OUTCOMES (RESULTS)
  ARE KNOWN BUT SPECIFIC OBSERVATIONS CANNOT BE
  KNOWN IN ADVANCE.
• EXAMPLES
• TOSS A COIN
• ROLL A DIE

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SAMPLE SPACE

• THE SET OF ALL POSSIBLE OUTCOMES OF A RANDOM
  EXPERIMENT IS CALLED THE SAMPLE SPACE.
• NOTATION S
• EXAMPLES
• FLIP A COIN THREE TIMES
• S

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EXAMPLE 2.

• AN EXPERIMENT CONSISTS OF FLIPPING A COIN AND
  THEN FLIPPING IT A SECOND TIME IF A HEAD OCCURS.
  OTHERWISE, ROLL A DIE.
• RANDOM VARIABLE
• THE OUTCOME OF AN EXPERIMENT IS CALLED A RANDOM
  VARIABLE. IT CAN ALSO BE DEFINED AS A QUANTITY
  THAT CAN TAKE ON DIFFERENT VALUES.

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EXAMPLE

• FLIP A COIN THREE TIMES. IF X DENOTES THE
  OUTCOMES OF THE THREE FLIPS, THEN X IS A RANDOM
  VARIABLE AND THE SAMPLE SPACE IS
• S HHH,HHT,HTH,THH,HTT,THT,TTH,TTT
• IF Y DENOTES THE NUMBER OF HEADS IN THREE FLIPS,
  THEN Y IS A RANDOM VARIABLE. Y 0, 1, 2, 3

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PROBABILITY DISTRIBUTION

• LET X BE A RANDOM VARIABLE WITH ASSOCIATED SAMPLE
  SPACE S. A PROBABILITY DISTRIBUTION (p. d.) FOR X
  IS A FUNCTION P WHOSE DOMAIN IS S, WHICH
  SATISFIES THE FOLLOWING TWO CONDITIONS
• 0 P (w) 1 FOR EVERY w IN S.
• P (S) 1, I.E. THE SUM OF P(S) IS
• ONE.

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REMARKS

• IF P (w) IS CLOSE TO ZERO, THEN THE OUTCOME w IS
  UNLIKELY TO OCCUR.
• IF P (w) IS CLOSE TO 1, THE OUTCOME w IS VERY
  LIKELY TO OCCUR.
• A PROBABILITY DISTRIBUTION MUST ASSIGN A
  PROBABILITY BETWEEN 0 AND 1 TO EACH OUTCOME.
• THE SUM OF THE PROBABILITY OF ALL OUTCOMES MUST
  BE EXACTLY 1.

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EXAMPLES

• A COIN IS WEIGHTED SO THAT HEADS IS TWICE AS
  LIKELY TO APPEAR AS TAILS. FIND P(T) AND P(H).
• 2. THREE STUDENTS A, B AND C ARE IN A SWIMMING
  RACE. A AND B HAVE THE SAME PROBABILITY OF
  WINNING AND EACH IS TWICE AS LIKELY TO WIN AS C.
  FIND THE PROBABILITY THAT B OR C WINS.

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EVENTS

• AN EVENT IS A SUBSET OF A SAMPLE SPACE, THAT IS,
  A COLLECTION OF OUTCOMES FROM THE SAMPLE SPACE.
• EVENTS ARE DENOTED BY UPPER CASE LETTERS, FOR
  EXAMPLE, A, B, C, D.
• LET E BE AN EVENT. THEN THE PROBABILITY OF E,
  DENOTED P(E), IS GIVEN BY

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FOR ANY EVENT E, 0 lt P(E) lt 1

• COMPUTATIONAL FORMULA
• LET E BE ANY EVENT AND S THE SAMPLE SPACE. THE
  PROBABILITY OF E, DENOTED P(E) IS COMPUTED AS

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EXAMPLES

1. A PAIR OF FAIR DICE IS TOSSED. FIND THE
   PROBABILITY THAT THE MAXIMUM OF THE TWO NUMBERS
   IS GREATER THAN 4.
2. ONE CARD IS SELECTED AT RANDOM FROM 50 CARDS
   NUMBERED 1 TO 50. FIND THE PROBABILITY THAT THE
   NUMBER ON THE CARD IS (I) DIVISIBLE BY 5, (II)
   PRIME, (III) ENDS IN THE DIGIT 2.

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NULL EVENT AN EVENT THAT HAS NO CHANCE OF
OCCURING. THE PROBABILITY OF A NULL EVENT IS
ZERO. P( NULL EVENT ) 0

• CERTAIN OR SURE EVENT AN EVENT THAT IS SURE TO
  OCCUR. THE PROBABILITY OF A SURE OR CERTAIN EVENT
  IS ONE.
• P(S) 1

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COMBINATION OF EVENTS

• INTERSECTION OF EVENTS
• THE INTERSECTION OF TWO EVENTS A AND B, DENOTED
• IS THE EVENT CONTAINING ALL
  ELEMENTS(OUTCOMES) THAT ARE COMMON TO A AND B.

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PICTURE DEMONSTRATION
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UNION OF EVENTS

• THE UNION OF TWO EVENTS A AND B, DENOTED,
• IS THE EVENT CONTAINING ALL THE ELEMENTS THAT
  BELONG TO A OR B OR BOTH.

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PICTURE DEMONSTRATION
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COMPLEMENT OF AN EVENT

• THE COMPLEMENT OF AN EVENT A WITH RESPECT TO S IS
  THE SUBSET OF ALL ELEMENTS(OUTCOMES) THAT ARE NOT
  IN A.
• NOTATION

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PICTURE DEMONSTRATION
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MUTUALLY EXCLUSIVE(DISJOINT) EVENTS

• TWO EVENTS A AND B ARE MUTUALLY
  EXCLUSIVE(DISJOINT) IF
• THAT IS, A AND B HAVE NO OUTCOMES IN COMMON.
• IF A AND B ARE DISJOINT(MUTUALLY EXCLUSIVE),

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PICTURE DEMONSTRATION
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ADDITION RULE

• IF A AND B ARE MUTUALLY EXCLUSIVE EVENTS, THEN
• GENERAL ADDITION RULE
• IF A AND B ARE ANY TWO EVENTS, THEN

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PICTURE DEMONSTRATION
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INDEPENDENCE OF EVENTS

• TWO EVENTS A AND B ARE SAID TO BE INDEPENDENT IF
  ANY OF THE FOLLOWING EQUIVALENT CONDITIONS ARE
  TRUE

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CLASSWORK EXAMPLES FROM PRACTICE EXERCISES SHEET
2

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CONDITIONAL PROBABILITY AND DECISION TREES

• LET A AND B BE ANY TWO EVENTS FROM A SAMPLE SPACE
  S FOR WHICH P(B) gt 0. THE CONDITIONAL PROBABILITY
  OF A GIVEN B, DENOTED
• IS GIVEN BY

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CLASSWORK EXAMPLES FROM PRACTICE EXERCISES SHEET
2
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GENERAL MULTIPLICATION RULE

• THE FORMULA FOR CONDITIONAL PROBABILITY CAN BE
  MANIPULATED ALGEBRAICALLY SO THAT THE JOINT
  PROBABILITY P(A and B) CAN BE DETERMINED FROM
  THE CONDITIONAL PROBABILITY OF AN EVENT. USING
• AND SOLVING FOR P(A and B), WE OBTAIN THE
  GENERAL MULTIPLICATION RULE

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CLASSWORK EXAMPLES FROM PRACTICE EXERCISES SHEET
2
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CONDITIONAL PROBABILITY CONTD

• CONDITIONAL PROBABLITY THROUGH BAYES FORMULA
• SHALL BE SKIPPED FOR THIS CLASS

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BAYES FORMULA FOR TWO EVENTS A AND B

• BY THE DEFINITION OF CONDITIONAL PROBABILITY,

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CLASSWORK EXAMPLES FROM PRACTICE EXERCISES SHEET
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