Independence
Two events A and B are said to be independent if any
following holds:
! P(A|B) = P(A)
! P(B|A) = P(B)
! P(A ∩ B) = P(A)P(B)
Otherwise, they are said to be dependent.
,Example
PCD / B) → A
Toss a fair coin two times
Let:
! A be the event that the two coins give the same resu
! B be the event that the first coin is a head.
Are the events A and B independent?
-
,Independence
The definition of independence extends to collection
two events.
A collection of n events A1 , . . . , An is said to be ind
every collection of k events Ai1 , . . . , Aik , k ≤ n, we
k
#
P Aij = P(Ai1 ∩ Ai2 ∩ . . . ∩ Aik
j=1
, Independence
In the case of n = 3 events—say, A, B and C —this
requires that
! P(A ∩ B) = P(A)P(B),
! P(A ∩ C ) = P(A)P(C ),
! P(B ∩ C ) = P(B)P(C ),
! P(A ∩ B ∩ C ) = P(A)P(B)P(C ).
It is possible for the first three conditions to hold wit
Two events A and B are said to be independent if any
following holds:
! P(A|B) = P(A)
! P(B|A) = P(B)
! P(A ∩ B) = P(A)P(B)
Otherwise, they are said to be dependent.
,Example
PCD / B) → A
Toss a fair coin two times
Let:
! A be the event that the two coins give the same resu
! B be the event that the first coin is a head.
Are the events A and B independent?
-
,Independence
The definition of independence extends to collection
two events.
A collection of n events A1 , . . . , An is said to be ind
every collection of k events Ai1 , . . . , Aik , k ≤ n, we
k
#
P Aij = P(Ai1 ∩ Ai2 ∩ . . . ∩ Aik
j=1
, Independence
In the case of n = 3 events—say, A, B and C —this
requires that
! P(A ∩ B) = P(A)P(B),
! P(A ∩ C ) = P(A)P(C ),
! P(B ∩ C ) = P(B)P(C ),
! P(A ∩ B ∩ C ) = P(A)P(B)P(C ).
It is possible for the first three conditions to hold wit
