Question

If a machine is correctly set up, it produces 80% acceptable items. If it is incorrectly set up, it produces only 30% acceptable items. From the past experience it was known that 90% of the setups are correctly done. If after a certain setup, the machine produces 2 acceptable items then the probability that the machine was correctly set up, is:

• A. \( 1/65 \)
• B. \( 72/75 \)
• C. \( 64/65 \)
• D. \( 3/75 \)

Hint:

Always check if events are independent when calculating joint probabilities like "2 acceptable items."

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We use Bayes' Theorem to find the posterior probability. Let \( C \) be the event of correct setup and \( I \) be the event of incorrect setup. Let \( E \) be the event that 2 items produced are acceptable.

Step 2: Key Formula or Approach:
\( P(C) = 0.9, P(I) = 0.1 \).
If setup is correct, \( P(E|C) = (0.8)^2 = 0.64 \).
If setup is incorrect, \( P(E|I) = (0.3)^2 = 0.09 \).

Step 3: Detailed Explanation:
By Bayes' Theorem:
\( P(C|E) = \frac{P(E|C)P(C)}{P(E|C)P(C) + P(E|I)P(I)} \)
\( P(C|E) = \frac{0.64 \times 0.9}{(0.64 \times 0.9) + (0.09 \times 0.1)} = \frac{0.576}{0.576 + 0.009} = \frac{0.576}{0.585} \)
Dividing both by 0.009:
\( \frac{64}{65} \).

Step 4: Final Answer:
The probability is \( 64/65 \).