Bayes Theorem

Study Notes – Bayes’ Theorem

Target Exam: IIT‑JEE (Main & Advanced) – Mathematics
Depth: Rigorous, exam‑oriented with derivations, key formulae, pitfalls, and a hard practice problem.

1. Quick Recap – Conditional Probability

Definition
For events (A,B) with (P(B) > 0):
[ P(A\mid B)=\frac{P(A\cap B)}{P(B)} ]

Product Rule
[ P(A\cap B)=P(A\mid B),P(B)=P(B\mid A),P(A) ]

2. Derivation of Bayes’ Theorem

From the product rule:

[ P(A\cap B)=P(A\mid B)P(B)=P(B\mid A)P(A) ]

Divide both sides by (P(B)) (non‑zero):

[ P(A\mid B)=\frac{P(B\mid A),P(A)}{P(B)} ]

Bayes’ Theorem (discrete case)
[ \boxed{P(A\mid B)=\frac{P(B\mid A),P(A)}{P(B)}} ]

3. Law of Total Probability (LTP)

If ({A_i}_{i=1}^n) form a partition of the sample space and (P(B)>0):

[ P(B)=\sum_{i=1}^{n}P(B\mid A_i),P(A_i) ]

Using LTP, Bayes’ Theorem can be written for multiple hypotheses:

[ P(A_k\mid B)=\frac{P(B\mid A_k)P(A_k)} {\displaystyle\sum_{i=1}^{n}P(B\mid A_i)P(A_i)} ]

4. Continuous Form (Density)

For continuous random variables (X,Y):

[ f_{X\mid Y}(x\mid y)=\frac{f_{Y\mid X}(y\mid x),f_X(x)}{f_Y(y)} ]

Here (f) denotes the probability density function (pdf).

5. Why Bayes Is Crucial for IIT‑JEE