Where does Bayes rule can be used?

Hint: In this question we use the conception of probability. Probability means the chance of an event happening or occurring.

Formula Used:>The Bayes rule is expressed in the following formula:
  First Formula: -
\[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B)}}\]
Where:
\[P(A|B)\] – the probability of event A occur, given event B has occurred.
\[P(B|A)\] – the probability of event B occur, given event A has occurred.
\[P(A)\] – the probability of event A.
\[P(B)\] – the probability of event B.
Here, Events A and B are independent events.
Second Formula: -
\[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B|{A^ - })P({A^ - }) + P(B|{A^ + })P({A^ + })}}\]
 Where:
\[P(B|{A^ - })\] – the probability of event B occur given that event A– has occurred.
\[P(B|{A^ + })\] – the probability of event B occur given that event A+ has occurred.
Here in the second formula of the Bayes rule , event A is a binary variable. The mutually exclusive events A– and A+ are outcomes of event A.



Complete step by step solution:Bayes theorem is applied to the questions where we have the probability of one event dependent on the second and some other partition on which these two are dependent and we need to find the probability of second dependent on first.
Besides statistics, the Bayes rule is also used in various disciplines, with medicine and pharmacology as the most notable exemplification.
 In addition, the rule is generally employed in different fields of finance.
Note: In these type of question, the general confusion of all students to apply the formula where what formula is need to apply . Student need to apply formula very carefully. If in question, Event A and B are independent events then we need to apply first formula \[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B)}}\] and if event A is binary variable then we need to apply the second formula\[\] \[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B|{A^ - })P({A^ - }) + P(B|{A^ + })P({A^ + })}}\]of Bayes rule.


Note: In this type of question, the general confusion of all students to apply the formula where what formula is needed to apply. Students need to apply formulas very carefully. If in question, Event A and B are independent events then we need to apply first formula \[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B)}}\] and if event A is binary variable then we need to apply the second formula\[\] \[P(A|B) = \dfrac{{P(B|A)P(A)}}{{P(B|{A^ - })P({A^ - }) + P(B|{A^ + })P({A^ + })}}\] of Bayes rule.

Additional Information:
In statistics and probability preposition, the Bayes Rule (also known as the Bayes Theorem) is a mathematical formula used to determine the tentative probability of events. Basically, the Bayes Rule describes the probability of an event Based on previous knowledge of the conditions that might be relevant to the event. Bayes rule formula discovered by Thomas Bayes in 1763. It considered the foundation of the special statistical inference approach that called Bayes inference.