Probability – Conditional & Bayes’ Theorem
WBJEE Mathematics – Key Study Notes
1. Quick Overview
| Concept | What it is | When to use it |
|---|---|---|
| Probability | Measure of likelihood of an event | Any problem with random outcomes |
| Conditional Probability | Probability of A given B has occurred | When events are linked (e.g., “given that a card is a heart”) |
| Bayes’ Theorem | Updating a prior probability after observing new evidence | Medical tests, quality control, urn problems, etc. |
Remember:
- All probabilities are between 0 and 1.
- In WBJEE, sample spaces are usually finite and equiprobable, unless stated otherwise.
2. Conditional Probability
2.1 Definition
For events (A) and (B) with (P(B) > 0),
[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} ]
- (P(A \mid B)) read as “probability of A given B”.
- (P(A \cap B)) is the probability that both events occur.
2.2 Key Properties
| Property | Symbol | Notes |
|---|---|---|
| Multiplication Rule | (P(A \cap B) = P(A \mid B),P(B)) | Used to compute joint probabilities |
| Independence | (P(A \mid B) = P(A)) iff (P(A \cap B) = P(A)P(B)) | Events have no influence on each other |
| Complementary | (P(A^c \mid B) = 1 - P(A \mid B)) | Useful when easier to compute the complement |
| Law of Total Probability | (P(B) = \sum_i P(B \mid A_i)P(A_i)) | When a sample space is partitioned into (A_i) |
2.3 Quick Formula Sheet
| Formula | Meaning |
|---|---|
| (P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}) | Conditional probability |
| (P(A \cap B) = P(A \mid B)P(B)) | Joint probability |
| (P(A \mid B^c) = \dfrac{P(A) - P(A \cap B)}{1 - P(B)}) | Conditional on complement |
| (P(B) = P(B \mid A)P(A) + P(B \mid A^c)P(A^c)) | Law of total probability |
3. Bayes’ Theorem
3.1 Definition
For events (A_1,\dots,A_n) forming a partition of the sample space and any event (B) with (P(B) > 0),
[ P(A_i \mid B) = \frac{P(B \mid A_i),P(A_i)}{\displaystyle \sum_{j=1}^{n} P(B \mid A_j),P(A_j)} ]
- Prior probability (P(A_i)) – initial belief about event (A_i).
- Likelihood (P(B \mid A_i)) – probability of observing evidence (B) if (A_i) is true.
- Posterior probability (P(A_i \mid B)) – updated belief after seeing (B).
3.2 Typical WBJEE Use‑Cases
| Scenario | Typical Term | Example |
|---|---|---|
| Medical testing | Disease prevalence | Probability of having a disease given a positive test |
| Quality control | Defective rate | Probability a defect is from line X given a defective product |
| Urn problems | Composition of urn | Probability the selected ball is red given a certain number of reds |
4. Worked Example 1 – Conditional Probability
Problem (WBJEE style)
A fair six‑sided die is rolled. Let
- (A =) “the outcome is an even number”
- (B =) “the outcome is greater than 3”
Find (P(A \mid B)).
Step‑by‑Step
| Step | Calculation | Result |
|---|---|---|
| 1. Sample space | ({1,2,3,4,5,6}) | – |
| 2. (B) outcomes | ({4,5,6}) | (P(B)=\dfrac{3}{6}=\dfrac{1}{2}) |
| 3. (A \cap B) outcomes | ({4,6}) | (P(A \cap B)=\dfrac{2}{6}=\dfrac{1}{3}) |
| 4. Apply definition | (P(A \mid B)=\dfrac{P(A \cap B)}{P(B)}\ |