Q.A manufacturer has three machine operators $A$, $B$ and $C$. The first operator $A$ produces 1% defective items whereas the other two operators $B$ and $C$ produce 5% and 7% defective items respectively. $A$ is on the job for 50% of the time, $B$ is on the job for 30% of the time and $C$ is on the job for 20% of the time. A defective item is produced. What is the probability that it was produced by $A$? OR Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values $x$ has the following form, where $k$ is a constant : $P(X=x) = \begin{cases} 0.1, & \text{if } x=0 \\ kx, & \text{if } x=1 \text{ or } 2 \\ k(5-x), & \text{if } x=3 \text{ or } 4 \\ 0, & \text{otherwise} \end{cases}$ Find the value of $k$. What is the probability that you study at least 2 hours, exactly 2 hours and at most 2 hours?

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. …