Question 1169525
Let's break down this problem using Bayes' Theorem.

**Define the Events**

* **C:** The machine is set up correctly.
* **G:** The first component produced is good.

**Given Probabilities**

* $P(C) = 0.85$ (Probability the machine is set up correctly)
* $P(C') = 1 - P(C) = 1 - 0.85 = 0.15$ (Probability the machine is not set up correctly)
* $P(G|C) = 0.96$ (Probability of a good component given the machine is set up correctly)
* $P(G|C') = 0.30$ (Probability of a good component given the machine is not set up correctly)

**We need to find:**

* $P(C|G)$ (Probability the machine is set up correctly given the component is good)

**Apply Bayes' Theorem**

Bayes' Theorem states:

$$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$

Where $P(G)$ can be found using the law of total probability:

$$P(G) = P(G|C) \cdot P(C) + P(G|C') \cdot P(C')$$

**Calculations**

1.  **Calculate $P(G)$:**
    $$P(G) = (0.96 \times 0.85) + (0.30 \times 0.15)$$
    $$P(G) = 0.816 + 0.045$$
    $$P(G) = 0.861$$

2.  **Calculate $P(C|G)$:**
    $$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$
    $$P(C|G) = \frac{0.96 \times 0.85}{0.861}$$
    $$P(C|G) = \frac{0.816}{0.861}$$
    $$P(C|G) \approx 0.9477$$

**Final Answer**

The probability that the machine is set up correctly given that the first component produced is good is approximately 0.9477.