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This lecture was delivered by Aatish Chippada at Alliance University for Statistics course. It includes: Sample, Space, Events, Collection, Outcomes, Relationship, Relative, Frequency, Occurance, Complementation
Typology: Slides
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ļµ^
Sample Space:- the collection of all possibleoutcome of an experiment. ļµ^
Event:- Any collection of outcomes for theexperiment (sub set of the sample space) ļµ^
Exp ā 30 year old woman lives to see her 70
th
birthday. ļµ^
Or event that some woman in diagnosed withcervical cancer before she reaches the age of 40.ā A particular plant (nuclear power plant) experiences a
melt down within the next 10 years
E
Let E, A, B be events, then(A&B) is the event that both A and B occur(AorB) is the even either A or B both occur (AUB)(not E) is the event that E does not occur
)
(^
B A^ ļ
) ( ) (^
A or cA
E
(A&B)
(AUB)
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ļµ^
Two or more events are said to be mutuallyexclusive if no two of them can occur, whenthe experiment is performed. i.e. no two ofthem have prob. That either of two will occurP(AUB)=P(A)+P(B)
Additive value of probability
P(A
UA 1
UA 2
3 ā¦UA
)=P(An
)+P(A 1
)+..P(A 2
)n
If events are not mutually Exclusive
P(AUB)=P(A)+P(B)-
)
(^
B A P^
ļ
ļµ^
The multiplicative Rule of Prob. States thatthe prob. That two events A and B will bothoccur is equal to the Prob. Of A multipliedby the Prob. Of B, given that A has alreadyoccurred. P(A&B)=P(A).P(B/A)
=P(A)P(B/A)
General Multiplication Rule
) =P(B)P(A/B)
(^
B A P^
ļ
)
(^
B A P^
ļ
) (
) ( ) / ( ) (
)
( ) / (^
B P
B A P B A P
similarly A P
B A P A B P^
ļ
ļ^
ļ½
ļ½
ļµ Primary useā Revise probabilities in accordance with
newly acquired informations. Suchrevised prob. Are conditionalprobabilities.
ļµ^
Exhaustive eventsā Events A
, A 1
, A 2
ā¦.A 3
are said to beK
exhaustive if at least one of them must occur.
Exp (E
, E 1
, E 2
) 3
D^
R^
I^
Governors
Events could be exhaustive +M. Exclusiveā If events are both exhaustive and mutually
exclusive then exactly one of them must occur.
mutually exclusive and exhaustive
The problem is to use these 6 prob. To determine the
conditional probabilities
P(A
/B), P(A 1
/B), P(A 2
/B) 3
Apply conditional probability ruleApply the general multiplication rule to the numerator and rule
of total prob. To the denominator P(B)=P(A
).P(B/A 1
)+P(A 1
).P(B/A 2
)+P(A 2
).P(B/A 3
) 3
So we get
Bayesās Rule
) )( ( )( ( )/ (^
2 2
2
BA BP P AB BP P B AP
ļ
ļ^
ļ½
ļ½
) / ( ). ( )
(^
2
2
2
A B P A P B A
P^
ļ½ ļ
) /( ).( ) /( ). ( )/ (). (
) /( ).(
)/ (
3 3 2 2 1 1
2 2
2
AB P AP AB P AP AB P AP
AB P AP
B AP
ļ«
ļ«
ļ½
ļ½^ k ļ„ļ½ ij
j j
i i
i
AB P AP
AB PA P BA P
) /( ).(
)/ (). ( )/ (
If an event E can happen in h ways out of total
ānā possible ways then the probability ofoccurrence of the event (its success) is The probability of non-occurrence (its failure)
is denoted by Then p+q=
1 & E
2 are two events, the probability
that E
2 occurs given that E
1 has occurred
is called āConditional probabilityā. It is denoted by Pr{E
/E 2
} or Pr{E 1
given 2
E^1
} Independent Events
: If the occurance or
non occurance of E
1 does not affect the
probability of occurance of E
, then 2
Pr{E
/E 2
} = Pr{E 1
} 2
E^1
& E
are independent events. 2
Compound events
:-If
both
occur, they are called compoundevents then Pr{E
}= Pr{E 2
Pr{E
2
are independent events.
Pr{E
}=Pr{E 2
} Pr{E 1
A ball is drawn at random from a box having 6
red balls, 5 blue balls and 4 green balls.What is probability that a ball drawn is (a)Red
(b) Green (c) Blue
(d) not red (e)
red or green Probability of red ball
4 5 6
6
sin
sin
} Pr{
ļ« ļ« ļ½
ļ½^
g
choo of ways total
ball reda g
choo of ways R
2 5 (^615) ļ½
Pr{B} Pr{G}
2 3
10 15 4 5 6
4 6
}
Pr{
ļ½
ļ½ ļ« ļ«
ļ«
ļ½
green or red
1 3
(^515)
4 5 6
5
ļ½ ļ½ ļ« ļ«
(^415) 4 5 6
4
ļ½ ļ« ļ«
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Two or more events are called
mutually exclusive if any of themāsoccurance excludes the occurrence ofthe others. If E
1
2
are mutually
exclusive events, Pr{E
In particular,Pr{E
}=Pr{E 2
}+Pr{E 1
}-Pr{E 2
set of values X
2
withk
respective probabilities p
, p 1
2
... p
.k
The function p(X) which has therespective values p
, p 1
2
... p . Fork
k^
is called the āprobability
functionā. Also called discrete prob.function
