BITSAT Mathematics – Study Notes
Topic: Bayes’ Theorem & Binomial Distribution
These notes are designed for quick revision and to give you a solid conceptual foundation for the BITSAT exam.
1. Bayes’ Theorem
1.1 Simple Explanation
Bayes’ Theorem tells us how to update our belief about an event after observing new evidence.
If we have two events A (hypothesis) and B (evidence), the probability that A is true given that B has occurred is:
[ P(A|B)=\frac{P(B|A),P(A)}{P(B)} ]
- (P(A)) – Prior probability of A.
- (P(B|A)) – Likelihood: probability of observing B if A is true.
- (P(B)) – Marginal probability of B (overall chance that evidence occurs).
- (P(A|B)) – Posterior probability: updated chance of A after seeing B.
1.2 Key Points
| Concept | What to remember |
|---|---|
| Conditional Probability | (P(A |
| Bayes’ Formula | (P(A |
| Independence | If (A) and (B) are independent, (P(A |
| Application | Medical testing, reliability, quality control, forensic analysis. |
| Common Pitfall | Mixing up (P(B |
1.3 Worked Example
Problem – A certain disease affects 2 % of a population.
A test for the disease gives a positive result with probability 90 % if the person has the disease (sensitivity = 0.90).
If the person does not have the disease, the test is still positive 5 % of the time (false‑positive rate = 0.05).
A randomly chosen person tests positive.
What is the probability that the person actually has the disease?
| Symbol | Meaning | Value |
|---|---|---|
| (D) | Person has the disease | 0.02 |
| (\bar D) | Person does not have the disease | 0.98 |
| (T^+) | Test is positive | – |
| (P(T^+ | D)) | Sensitivity |
| (P(T^+ | \bar D)) | False‑positive rate |
Step 1 – Compute (P(T^+)) (marginal probability of a positive test).
[
P(T^+)=P(T^+|D)P(D)+P(T^+|\bar D)P(\bar D)
=0.90(0.02)+0.05(0.98)=0.018+0.049=0.067
]
Step 2 – Apply Bayes’ Theorem.
[
P(D|T^+)=\frac{P(T^+|D)P(D)}{P(T^+)}
=\frac{0.90\times0.02}{0.067}\approx\frac{0.018}{0.067}\approx0.269
]
Answer: About 26.9 % chance the person actually has the disease.
Insight – Even with a highly accurate test, the low prevalence (prior) keeps the posterior probability modest.
1.4 Quick Revision Checklist
- Write down the formula.
- Identify prior, likelihood, and evidence.
- Compute marginal probability (denominator) correctly.
- Check units/percentages.
- Verify that probabilities lie between 0 and 1.
2. Binomial Distribution
2.1 Simple Explanation
The Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same success probability.
- (n) – Number of trials
- (p) – Probability of success in a single trial
- (k) – Number of successes observed
The probability of exactly (k) successes is
[ P(X=k)=\binom{n}{k}p^{,k}(1-p)^{,n-k} ]
where (\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}).
2.2 Key Points
| Property | Formula / Info |
|---|---|
| Mean | (\mu = np) |
| Variance | (\sigma^2 = np(1-p)) |
| Standard Deviation | (\sigma = \sqrt{np(1-p)}) |
| Symmetry | If (p=0.5) it’s symmetric; otherwise skewed. |
| Applications | Coin tosses, quality control, survey responses, etc. |
| Special Cases |
- (p=0.5): Binomial equals a symmetric distribution.
- (n=1): reduces to a Bernoulli distribution. |
| Approximation | For large (n) and (np(1-p)) large, use Normal approximation:
[ X \approx N(np,;np(1-p)) ] |
2.3 Worked Example
Problem – A machine produces light bulbs, and 3 % of them are defective.
If you randomly pick 10 bulbs, what