Questions: were found to be defective, what is the :probability that it was made by machine C a. 0.2514 b. 0.3901 c. 0.5510 D.0.2041

Transcript text: were found to be defective, what is the :probability that it was made by machine C a. 0.2514 b. 0.3901 c. 0.5510 D.0.2041

Solution

Solution Steps

To solve this problem, we need to use Bayes' Theorem to find the probability that a defective item was made by machine C. We need to know the probability of a defective item being produced by each machine and the overall probability of an item being defective. Then, we can apply Bayes' Theorem to find the desired probability.

Step 1: Define the Probabilities

Let:

\( P(C) = 0.3 \): Probability that an item is made by machine C.
\( P(D|C) = 0.05 \): Probability that an item is defective given it is made by machine C.
\( P(A) = 0.4 \): Probability that an item is made by machine A.
\( P(B) = 0.3 \): Probability that an item is made by machine B.
\( P(D|A) = 0.02 \): Probability that an item is defective given it is made by machine A.
\( P(D|B) = 0.03 \): Probability that an item is defective given it is made by machine B.

Step 2: Calculate Total Probability of Defectiveness

The total probability of an item being defective, \( P(D) \), is calculated as follows: \[ P(D) = P(A) \cdot P(D|A) + P(B) \cdot P(D|B) + P(C) \cdot P(D|C) \] Substituting the values: \[ P(D) = 0.4 \cdot 0.02 + 0.3 \cdot 0.03 + 0.3 \cdot 0.05 = 0.008 + 0.009 + 0.015 = 0.032 \]

Step 3: Apply Bayes' Theorem

To find the probability that a defective item was made by machine C, \( P(C|D) \), we use Bayes' Theorem: \[ P(C|D) = \frac{P(D|C) \cdot P(C)}{P(D)} \] Substituting the known values: \[ P(C|D) = \frac{0.05 \cdot 0.3}{0.032} = \frac{0.015}{0.032} = 0.46875 \]