Q.If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup.

Concept: Conditional Probability (Bayes' Theorem)

Let $C$ = correctly set up, $A$ = produces an acceptable item.

Given: $P(C) = 0.8$, $P(A|C) = 0.9$, $P(A|C') = 0.4$.

We need $P(C \mid \text{2 acceptable items})$. Since items are independent given the setup:

1. $P(\text{2 acceptable} \mid C) = (0.9)^2 = 0.81$

   $P(\text{2 acceptable} \mid C') = (0.4)^2 = 0.16$

2. $P(\text{2 acceptable}) = P(C) \cdot 0.81 + P(C') \cdot 0.16$

   $= 0.8 \times 0.81 + 0.2 \times 0.16 = 0.648 + 0.032 = 0.68$

3. By Bayes' theorem: …

Let $C$ = correct setup, $C'$ = incorrect setup, and $E$ = "two acceptable items".

Items are independent given the setup, so

Total probability of the observation:

$P(E)=(0.81)(0.8)+(0.16)(0.2)=0.648+0.032=0.68.$ …

Method: Bayes' Theorem (finding the cause from the observed outcome)

Use this when you observe repeated results from one of two hidden states and want the probability of the state given the results. You know the per-item success rate for each setup, and want $P(\text{correct setup}\mid\text{observed items})$.

Steps

Step 1: Set up the partition of possible causes.

List the mutually exclusive, exhaustive hypotheses (here: the machine was set up correctly or incorrectly) and write their prior probabilities $P(E_i)$ (these must sum to $1$).

Step 2: Write each likelihood.

For each cause, state $P(A\mid E_i)$ — the probability of the observed outcome (here: the machine produces two acceptable items) under that cause.

Step 3: Get the total probability of the outcome (the denominator).

By the law of total probability,

Step 4: Apply Bayes' theorem for the cause you want.

$P(E_k\mid A)=\frac{P(E_k)\,P(A\mid E_k)}{\sum_i P(E_i)\,P(A\mid E_i)}.$ …

Common Mistakes

Mistake 1: Using the single-item rate $0.9$ instead of squaring it.

Why it's wrong: the observation is two acceptable items, so the likelihood is $(0.9)^2=0.81$ for a correct setup (and $(0.4)^2=0.16$ for an incorrect one). Correct approach: square the per-item probabilities before applying Bayes' theorem.

Mistake 2: Omitting the incorrect-setup branch from the denominator.

Why it's wrong: two acceptable items can also come from an incorrectly set-up machine. Correct approach: $P(\text{2 acceptable})=P(C)(0.9)^2+P(C')(0.4)^2=0.68$. …