Q.A factory has two machines I and II. Machine I produces 40% of items of the output and Machine II produces 60% of the items. Further 4% of items produced by Machine I are defective and 5% produced by Machine II are defective. An item is drawn at random. If the drawn item is defective, find the probability that it was produced by Machine II. OR If $y=(\cos^{-1}x)^2$ prove that $(1-x^2)\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} - 2 = 0$, hence find $y_2$ when $x=0$.

Bayes' Theorem: Learning from Evidence

Imagine you have a bag with 3 red marbles and 7 blue marbles. If you pick one at random, the chance it's red is 3 out of 10 — that's straightforward. But now suppose someone picks a marble, doesn't show it to you, but tells you it's not blue. Suddenly, the only possibilities left are the red marbles. The probability that the hidden marble is red jumps to 1 (certainty). You just updated your belief based on new evidence.

That's the core idea of Bayes' Theorem: how to revise a probability when you get new information. It answers the question: Given that I now know B happened, how should I change my belief about A?

The Intuition in One Sentence

Bayes' Theorem says: The probability that A is true, given that B is true, equals the probability that B would happen if A were true, times the original probability of A, divided by the overall probability of B.

In other words: Your updated belief = (likelihood of the evidence under your hypothesis) × (your prior belief) / (total probability of the evidence).

The Precise Statement

Let $A$ and $B$ be two events. Then:

Where:

• $P(A \mid B)$ is the posterior probability — what you want: the probability of $A$ given that $B$ occurred.
• $P(B \mid A)$ is the likelihood — how probable the evidence $B$ is if $A$ is true.
• $P(A)$ is the prior probability — your initial belief about $A$ before seeing any evidence.
• $P(B)$ is the marginal probability of $B$ — the total chance that $B$ happens, regardless of $A$.

Why It Works: A Simple Example

Suppose 1% of a population has a disease. A test for the disease is 99% accurate: it correctly identifies 99% of those who have it (true positive) and correctly says 99% of those who don't have it are negative (true negative). You take the test and get a positive result. What is the probability you actually have the disease?

Many people guess 99%. But that's wrong — and Bayes' Theorem shows why.

Let $D$ = "has the disease", $T^+$ = "tests positive". We know:

• $P(D) = 0.01$ (prior)
• $P(T^+ \mid D) = 0.99$ (likelihood)
• $P(T^+ \mid \text{not } D) = 0.01$ (false positive rate)

First, find $P(T^+)$, the total probability of a positive test:

Now apply Bayes:

So even with a positive test, there's only a 50% chance you have the disease. The test is good, but the disease is rare — most positive results come from the large number of healthy people who get false positives. …