State and prove Baye's Theorem ?
Answers
Answer:
If A and B are two independent events, then the probability of event A when B is true is calculated as the ratio of probability of event B such that event A is true and the individual probability of event A to the individual probability of event B. Bayes theorem Formula is written as:
P(A/B)=P(B/A).P(A)P(B)
In the above mentioned Bayes Theorem Formula,
P (A | B) is the probability of event A being true when event B is true.
P (B | A) is the probability of event B being true when event A is true.
P (A) is the probability of event A being true.
P (B) is the probability of event B being true.
An Important Concept Required to State and Prove Bayes Theorem:
Conditional probability is the probability of one event when one or more other individual events are true. It can be better explained with the help of an example.
Suresh visits a library in which one of the book racks contains 3 rows. All the three rows are stacked with a mixture of reference books, journals and annual reports. Let us consider that Suresh picks a book from the second rack. The probability of whether the book picked by Suresh is a reference material depends on the other two events. (i.e. whether the book is a journal or an annual magazine). In general, conditional probability means the measure of probability of one event when the other event is true.
P(A/B)=P(A∩B)P(B)
where , A and B are two individual events and P (B) not equal to zero.
P (A | B) is the probability of event A being true when event B is true.
P (A ⋂ B) is the probability of occurrence of both A and B.
P (B) is the probability of individual event B.
How to State and Prove Bayes Theorem:
Bayes theorem formula is stated as
P(A/B)=P(B/A).P(A)P(B)
Bayes theorem proof can be derived using the concept of conditional probability. The probability of occurrence of both the events A and B is given in terms of their individual probabilities and conditional probability as:
P (A ∩ B) = P (A). P (B | A)
Similarly the occurrence of both the events simultaneously can also be given in terms of the probability of second event as:
P (A ∩ B) = P (B). P (A | B)
In both the equations, the left hand side is equal. So RHS can be equated.
P (B). P (A | B) = P (A). P (B | A)
Further simplification gives the Bayes theorem formula as
P(A/B)=P(B/A).P(A)P(B)
Step-by-step explanation:
Answer:
Baye's theorem states that the conditional probability of an event based on the occurrence of another event, is equal to the likelihood of the second event given the first event multiplied by the probability of first event.