CONDITIONAL PROBABILITY

1
CONDITIONAL PROBABILITY
Consider two events, A and B.
Suppose we know that B has occurred.
This knowledge may change the probability that A
will occur.

We denote by P(AB) the conditional probability
of event A given that B has occurred.
2
To obtain a formula for P(AB), let us refer to
the following figure
3
Note that the knowledge that B has occurred
effectively reduces the sample space from S to B.
Therefore, interpreting probability as the area,
P(AB) is the proportion of the area of B
occupied by A
4
Example- Tossing Two Dice Conditional Probability
An experiment consists of tossing two fair dice
which has a sample space of 6x636 outcomes.
Consider two events ASum of dice is 4 or 8
and BSum of dice is even
5
The sum of 4 or 8 can be achieved in eight ways
with two dice, so A consists of the following
elements A(1,3), (2,2), (3,1), (2,6), (3,5),
(4,4), (5,3), (6,2)
The sum of the dice is even when both have either
even or odd outcomes, so B contains the following
pairs B(1,1), (1,3), (1,5), (3,1), (3,3),
(3,5), (5,1), (5,3), (5,5), (2,2), (2,4),
(2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)
6
Thus A consists of 8 outcomes, while B consists
of 18 outcomes A(1,3), (2,2), (3,1), (2,6),
(3,5), (4,4), (5,3), (6,2)
B(1,1), (1,3), (1,5), (3,1), (3,3), (3,5),
(5,1), (5,3), (5,5), (2,2), (2,4), (2,6),
(4,2), (4,4), (4,6), (6,2), (6,4), (6,6)
Furthermore, A is a subset of B.
7
Assuming that all outcomes are equally likely,
the conditional probability of A given B is