Conditional Probability
When we discuss conditional probability, we need to understand two important concepts:

If A and B are events, then

1. A B
signifies "either the event A or the event B or both."

2. A B
signifies "both the event A and the event B "

To illustrate these concepts let us refer to the experiment of tossing a coin twice.
Let

A be the event "at least one head occurs"

A = { HT, TH, HH } ,

and let

B be the event "the second toss results in a tail."

B = { HT, TT }

and so we have:

A B = { HT, TH, HH, TT }

and

A B = { HT }

Definition of Conditional Probability:

Let A and B be two events such that P(A) > 0, and let P(B | A) be the probability of B given that A has occurred. Since A is known to have occurred, we can define:

P ( A B )

P ( B | A ) =

P ( A )

Example problem in conditional probability:

What is the probability that a single toss of a die results in a number less than 4 if

(a) no other information is given,
(b) it is given that the toss resulted in an odd number.