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Math 103Statistics andProbability
Probability
CJD
Notation
P

denotes a probabilityA, B, ...

denote specific eventsP (A)

denotes the probabilityof event A occurringP (A and B) – denotes the probabilityof event A and event B occuringsimultaneouslyP (A or B) – denotes the probability thatevent A occurs or event B occurs(or both).
CJD
Classical Probability
Method 1:
Classical Approach
If a procedure has
n
different simple events, each with an equalchance of occurring, and
s
is the number of ways event A canoccur, then
P(A)
=
number of ways A can occurnumber of differentsimple events
sn
=
Equally Likely Simple EventsIf there are
n
simple events in the sample space andthey are all equally likely, then the probability of theoccurrence of each one is
1/
n
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Empirical Probability
Method 2: Experimental Approach
Conduct (or observe) an experiment a large number of times, and count thenumber of times event A actually occurs, then an estimate of P(A) is
P(A)
=
number of times A occurrednumber of times trial was repeatedLaw of Large Numbers:
As a procedure is repeated again and again, theexperimental probability of an event tends toapproach the actual probability. Error is about
n
1
2
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Classical vs Empirical
The classical approach is the actual probability.The relative frequency approach is an approximation.
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Subjective Probability
Method 3:
Subjective Approach
P(A), the probability of A, is found by simply guessing or estimatingits value based on knowledge of the relevant circumstances.
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Probability Limits
ã The probability of an impossible event is
0
.ã The probability of an event certain to occur is
1
.ã
0
≤≤≤≤
P(A)
≤≤≤≤
1
CertainLikely5050 ChanceUnlikelyImpossible10.50
ãThe
total
of theprobabilities of allsample points in asample space
must equal 1
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Computing Probabilities
Step 1 :
Determine the sample space in the problem.
Step 2 :
Determine the sample pointsfor the event being considered
Step 3 :
Assign probabilities to the sample points
Step 4 :
Determine applicable probability formulasto compute the desired probabilityIf the event is complicated, break it down intoseveral “simpler” events.
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Example
960,598,2
552
=
What is the probability that a 5card hand from a standard52card deck is a flush ?Number of ways to have a 5card hand =Number of ways to have a flush =
148,5287,14
5134
=⋅=
⋅
Probability of a flush = 5148/2598960 = 0.00198What is the probability that a 5card hand from a standard52card deck is a royal flush ?Probability of a royal flush = 4/2598960 = 0.00000154
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Complementary Events
P(A)
The complement of event A, denoted by A,consists of all outcomes in which event A does
not
occur.
P(A)
Note: Sometimes, instead of A, the notation
A’
is used.
P(A) + P(A’) = 1
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Example
Example: Testing CorollasToyota wants to conduct a test of its new Corolla model. A pool of 50drivers has been recruited, 20 or whom are men. When the firstperson is selected from this pool,
(a) what is the probability of getting a man?
(b) What is the probability of getting a woman?P(selecting a man) = num of men drivers / num of drivers= 20 / 50 = 0.4P(selecting a woman) = num of women drivers / num of drivers= 30 / 50 = 0.6OR because selecting a man and selecting a woman arecomplementary events,P(selecting a woman) =
1 – 0.4
= 0.6
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Another Example
Find the probability that a couple with 3 children has at least 1girl.
Assume a child is equally likely to be a boy as to be a girl.
If P(A) = P(getting at least 1 girl), thenP(A) = 1  P(A’)where P(A’) is P(getting no girls)P(A’) = (0.5)(0.5)(0.5) = 0.125P(A) = 1  0.125 = 0.875
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Odds
Odds in favor
of event A is thereciprocal of the odds against that event,b:a (or ‘b to a’)
Odds against
event A occurring is theratio P(A’) / P(A), usuallyexpressed in the form of a:b(or ‘a to b’), where
a
and
b
areintegers with no common factors
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Odds Example
Example: Testing CorollasToyota wants to conduct a test of its new Corollamodel. A pool of 50 drivers has been recruited, 20or whom are men.
ã The odds against selecting a man is 3:2ã The odds in favor of selecting a man is 2:3
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Probability of Compound Events
In general,
P(A or B) = P(A) + P(B)  P(A and B)
If A and B are mutually exclusive events,Then P(A and B) = 0So,
P(A or B) = P(A) + P(B)
CJD
Men Women Boys Girls TotalsSurvived 332 318 29 27 706Died 1360 104 35 18 1517Total 1692 422 64 56 2223
Example
Find the probability of randomly selecting a man or a boy.P(man or boy) = 1692 + 64 = 1756 = 0.7902223 2223 2223
* Mutually Exclusive *
Survival data after a plague on a small town :