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The updated chance of an event occurring after additional information is taken into account is known as posterior probability.

If for two events A and B stands

\({\rm{P(A}}\mid {\rm{B) = P(A)}}\)we say that they are independent. They are dependent otherwise.

Conditional probability of A given that the event B has occurred, for which\({\rm{P(B) > 0}}\), is

\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)for any two events \({\rm{A}}\) and \({\rm{B}}\).

We want to prove that \({A^\prime }\) and B are independent by the definition. Using definition of conditional probability, we have

\(\begin{array}{c}P\left( {{A^\prime }\mid B} \right) &=& \frac{{P\left( {{A^\prime } \cap B} \right)}}{{P(B)}}\\ &=& \frac{{P(B) - P(A \cap B)}}{{P(B)}}.........(1)\\ &=& \frac{{P(B) - P(A)P(B)}}{{P(B)}}........(2)\\ &=& \frac{{P(B)(1 - P(A))}}{{P(B)}}\\ &=& 1 - P(A)\\ &=& P\left( {{A^\prime }} \right) ...............(3)\end{array}\)

(1): the following is true

\(\begin{array}{c}P(B) = P\left( {\left( {{A^\prime } \cap B} \right) \cup (A \cap B)} \right)\\ = P\left( {{A^\prime } \cap B} \right) + P(A \cap B)\end{array}\)

from which we have what we need

\(P\left( {{A^\prime } \cap B} \right) = P(B) - P(A \cap B)\)

(2): we use multiplication property (we can because we assume that A and B are independent,

(3): \(P(C) + P\left( {{C^\prime }} \right) = 1\)for any event\({\rm{C}}\).

Multiplication Property: Two events A and B are independent if and only if

\(P(A \cap B) = P(A) \cdot P(B)\)

From

\(P\left( {{A^\prime }\mid B} \right) = P\left( {{A^\prime }} \right)\)

and definition given above, events \({A^\prime }\)and \({\rm{B}}\) are independent.