Question

Amit wants to use Bayes’ formula $P\left(\frac{Bi}{A}\right)=\frac{P\left(Bi\right)P\left(\frac{A}{Bi}\right)}{\sum \left(Bi\right)P\left(\frac{A}{Bi}\right)}$
To use this formula,

• A. $B_1,B_2\cdots B_n$ should be independent
  
• B. $B_1,B_2\cdots B_n$ should be exhaustive
• C. $B_1,B_2\cdots B_n$ should be disjoint
• D. $A,B_1,B_2\cdots B_n$ should be disjoint
• E. $A\cap B_i=\varphi forall{B}_{i}s$

Answer 1

Answer: (B), (C)

$B_1,B_2\cdots B_n$ should be disjoint

Let’s first see what Bayes’ theorem says and then check which all statements are consistent with it. If an event A can occur with one of the n mutually exclusive and exhaustive events $B_1,B_2\dots ..B_n$ and the probabilities $P\left(\frac{A}{B_1}\right),P\left(\frac{A}{B_2}\right)\cdots P\left(\frac{A}{B_n}\right)$ are known, then $P\left(\frac{Bi}{A}\right)=\frac{P\left(Bi\right)P\left(\frac{A}{Bi}\right)}{\sum P\left(Bi\right)P\left(\frac{A}{Bi}\right)}$
So the conditions on Bi s are
1) They should be disjoint
2) They should be exhaustive
$\Rightarrow$ options B and C are correct.