Probability - 2

Venn diagram :-
I A (A but not B )

II A B

III B ( B but not A

IV
P(A or B) = P(at least one of A or B)
P(A img B) = P(A) + P(B) - P(A img B)
= I + II +III
= (I + II) + (III + II) - II
P(A B) = P(A) + P(B) - P (A B)
P (A B) = P ( A ) + ( B ) + P (A B )

= P(A) + P( B)

=| - P ( )


P (exactly one A or B) = I + III = P ()

=P(A) + P(B) - 2P (A B)

= P(A B ) - 2P(a B)
Note :- For mutually exclusive event

P(A B) = 0
For exhaustive event
(A img B) = Sample space
Conditional probability :-

Probability of ocurence of an event B when it is known that some event A has already occured .It is P(B/A)

P(B/A) =

P(B/A) = , 0 P(B) 1

P(B A) =

P(B/A) = P(A )

Note - For independent events -
P(A B) = P(A). P(B)
Dumb Question :- How P(A B) = P(A) P(A/B)
Ans - P(A/B) = probabilituy of ocurence of A in B
ie, simultanous occurence
Probability of simultaneous occurence of A and B
P(A/B) = Probability of B



P(A/B) =

Illustration - If

Find

Ans -





P(A/B) =



=



=

P(A / B) + = 1 = 1

= 1 - P(A / B)

=

=

Bernoulli strial :- An experiment out come



p + q = 1

P(getting r' s success) = n C_r p^r q^n-r

P

= p^r q^n-r nC_r img arrangement in a row

P (getting r + 1 success) = nC_r+1 q^n-r-1p ^r+1