Question

State and prove Bayes theorem

Answer 1

Answer:

In probability theory and statistics, Bayes’s theorem (alternatively Bayes’s law or Bayes’s rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event[1]. For example, if the probability that someone has cancer is related to their age, using Bayes’ theorem the age can be used to more accurately assess the probability of cancer than can be done without knowledge of the age.

Step-by-step explanation:

Let A and B be two events and let P(A|B) be the conditional probability of A given that B has occurred. Then Bayes' theorem states that:

P(B|A)=P(A|B)P(B)P(A)=P(A|B)P(B)P(A|B)P(B)+P(A|Bc)P(Bc)(1)

In other words, Bayes' theorem gives us the conditional probability of B given that A has occurred as long as we know P(A|B), P(A|Bc) and P(B)=1−P(Bc).

The proof of this equation is quite simple. First, consider the following facts:

P(A|B):=P(A∩B)P(B)⟹P(A∩B)=P(A|B)P(B)(2)

P(A)===P((A∩B)∪(A∩Bc))P(A∩B)+P(A∩Bc)P(A|B)P(B)+P(A|Bc)P(Bc)(3)

where on equation (3) the fact that A∩B and A∩Bc are mutually exclusive events was used. Therefore, since A∩B=B∩A:

P(B|A)===P(B∩A)P(A)P(A|B)P(B)P(A)P(A|B)P(B)P(A|B)P(B)+P(A|Bc)P(Bc)(4)

as we wanted to prove. We can also prove a very interesting formula using Bayes' theorem. From equation (4) (used with both B and Bc) and equation (3), we have:

P(B|A)+P(Bc|A)=P(A|B)P(B)P(A)+P(A|Bc)P(Bc)P(A)=P(A)P(A)=1(5)

so we obtain:

P(B|A)=1−P(Bc|A)(6)