Question

Why is Bayes rule so important?

Answer 1

Importance of Bayes rule

Bayes theorem allows us to reverse a conditional probability.

Thus, suppose we know the chance of testing positive for cancer, given we have cancer.
If we actually tested positive for cancer, though, we’d want to know: what are the chances I have cancer, given I tested positive? Bayes theorem shows us how to mathematically reverse this probability to find the answer to this question.

Thus, The Bayes rule help us to find the conditional probability of an event based on prior knowledge of conditions that might be related to the event.

Let us consider an example,

In two bags, black and white balls are kept. Bag $1$ has $4$ white and $3$ black and Bag $2$ have $5$ white and $2$ black. One ball is drawn, what is the conditional probability of the selected ball being a black ball from bag$1$?
Using Bayes Theorem

Let $A$ be an event that the Ball is Black.

Let $R_1$ be an event that chosen ball is drawn from Bag $1$

Let $R_2$ be an event that chosen ball is drawn from Bag $2$

$\therefore P\left(R_1\right)=P\left(R_2\right)=\frac{1}{2}$

$P\left(A\right)=P\left(A\right|R_1)+P(A\left|R_2\right)$

Where,

$P\left(A\right|R_1)=$Probability that drawn ball is black and is from Bag $1$

$P\left(A\right|R_2)=$Probability that drawn ball is black and is from Bag $2$

$\Rightarrow P\left(A\right)=\frac{1}{2}\times \frac{3}{7}+\frac{1}{2}\times \frac{2}{7}$

$\Rightarrow P\left(A\right)=\frac{3}{14}+\frac{2}{14}$

$\Rightarrow P\left(A\right)=\frac{5}{14}$

$P\left(R_1\right|A)=\frac{P\left(R_1\right)P\left(A\right|R_1)}{P\left(A\right)}$

Where

$P\left(R_1\right|A)=$Probability that drawn ball is from bag $1$ being it is given that the ball is black.

$\Rightarrow P\left(R_{1}A\right)=\frac{\frac{1}{2}\times \frac{3}{7}}{\frac{5}{14}}$

$\Rightarrow P\left(R_{1}A\right)=\frac{\frac{3}{14}}{\frac{5}{14}}$

$\Rightarrow P\left(R_{1}A\right)=\frac{3}{5}$