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The Definition Of Conditional
Probabilities
The conditional probabilities of
an event B given event A is the
probability that B will occur
given that A has occurred, and
is a probability base on a
reduce sample space.
The Formula Of Conditional
Probability
The conditional probability of event B
given event A is defined as
P(A  B)
P(B|A) = P(A)
provided that P(A) >0.
(i)
P(A B) is the probability that event A and
event B happen together.
(ii) P(B|A) is the probability that event B
occurs that event A occurs, event A is the
condition.
P(B|A) ≠ P(A|B)
Two fair dice are rolled. Find the
probability that
a)both numbers turned up are the same,given that the
sum of the numbers is greater than 6.
Answer:The required probability = P(A|B)
=
P( A  B)
P( B)
=
3 36

36 21
=
1
7
b)The sum of the numbers turned up is
greater than 6,given that both numbers
are the same.
Answer:The required probability=P(B|A)
=
P( A  B)
P ( A)
=
3 36

36 6
=
1
2
Three truncated octahedron (截角正八面體) are
rolled. One is black and two are red.
Find the probability that
a) the dice is black,
given that the number is ‘5’.
Answer: The required probability = P(black|’5’)
=
P (black '5' )
P ('5' )
=
1 1
 8
3 8
=
1
3
b) The number is ‘8’, given that the dice is red.
Answer: The required probability = P(‘8’|red)
=
P('8'red )
P(red )
=
1 2 3
 
8 3 2
=
1
8
c)the number is even,given that the dice is red.
Answer:The required probability =P(even|red)
=
P(even  red )
P(red )
=
4 2 3
 
8 3 2
=
1
2