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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 EXPERIMENTAN 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
1. FLIP A COIN THREE TIMES
S =
5
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:
1. 0 ≤ P (w) ≤ 1 FOR EVERY w IN S.
2. 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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EXAMPLES1. 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
n
iiwPEP
1
)()(
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FOR ANY EVENT E, 0 < P(E) < 1
• COMPUTATIONAL FORMULA
• LET E BE ANY EVENT AND S THE SAMPLE SPACE. THE PROBABILITY OF E, DENOTED P(E) IS COMPUTED AS
)(
)()(
Sn
EnEP
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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.
BA
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.
BA
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: 'AORAC
)(1)( APAP C
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),
BA
0)()( PBAP
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
)()()( BPAPBAP
)()()()( BAPBPAPBAP
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:
)()( APBAP
)()( BPABP
)().()( BPAPBAP
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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) > 0. THE CONDITIONAL
PROBABILITY OF A GIVEN B, DENOTED
IS GIVEN BY
)( BAP
)(
)()(
BP
BAPBAP
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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
)(
)()(
BP
BandAPBAP
)().()( BPBAPBandAP
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CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2
CONDITIONAL PROBABILITY CONT’D
• CONDITIONAL PROBABLITY THROUGH BAYE’S 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,
)(
)().(
)(
)()(
AP
BPBAP
AP
BandAPABP
)().()().(
)().(CC BPBAPBPBAP
BPBAP
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CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2
