Baye’s theorem is a very interesting topic of Statistics and Probability Theory. The Baye’s theorem is an alternative of Baye’s Law or Rules. |

**Example 1:**A company has two plants to manufacture scooters. Plant 1 manufacture 70% of the scooters and plant 2 manufacture 30%. At plant 1, 80% of the scooters are rated as of

standard quality and at plant 2, 90% of the scooters are rated as of standard

quality. What is the probability that it has come from plant 2?

**Solution:**

Let $E_1$, $E_2$

and A be the following events.

$E_1$ = Plant 1

is chosen,

$E_2$ = Plant 2

is chosen, and

A = Scooter is of standard

quality.

P($E_1$) =$\frac{70}{100}$ , P($E_2$) = $\frac{30}{100}$ , P(A | E1) = $\frac{80}{100}$ , P(A|E2) = $\frac{90}{100}$

We are required to find

By Bayes theorem, we

have

P($E_2$ | A)= $\frac{P(E_2)P(A|E_2)}{P(E_1)P(A|E_1)+ P(E_2)P(A|E_2)}$

P($E_2$ | A) = $\frac{\frac{30}{100}\frac{90}{100}}{\frac{70}{100} \frac{80}{100}+ \frac{30}{100}\frac{90}{100}}$

=$\frac{27}{56+27}$

= $\frac{27}{83}$

**Application of Bayes theorem:**

- Bayes theorem is used to determine probability of an event occurring.
- Bayes helps us to estimate prior probabilities of the events we are concerned with.
- Using Bayes rule we can predict the outcome of an event like sporting event, political elections, etc.

**1)**Bayesian Interpretation: This interpretation is based on the Baye’s law. In this the probability measures the degree of beliefs. Bays theorem joins the degree of belief in a proposition before and after accounting the given evidences. For instance suppose that person A proposes that the biased coin is twice as likely to land heads than tails. Now the degree of belief in this can be true initially is 50%. The coin is now flipped many times in order to collect all the evidences. So the belief may rise to 70% if all the evidences support the proposition.

For proposition G and evidence H:

**a)**P (G), the prior is the initial degree of the belief in proposition G.

**b)**P (G|H) the posterior is the degree of belief which is accounted for the evidence H.

**c)**P (B|A) / P (B) represents the support evidence H provides for proposition G.

**2)**Frequentist Interpretation: This is defined with respect to a great number of trials where each trial produces one outcome from a given Set of all the possible outcomes. An event is basically a subset of the set of all possible outcomes. The probability f an event G is P (G) is the proportion of all the trials which produce an outcome in event G. Similarly the probability of H is P (H). Now here if only trials in which event G occurs are considered, the proportion in which event H also occurs is P (H|G). Similarly if only trials in which event H occurs are considered, the proportion in which event H also occurs is P (G|H). So it completely relates to the inverse or just opposite representations of the probabilities which concern with two events.