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Conditional probability allows us to determine the probability of an event occurring, given that another event has occurred. In the context of the given problem, we are looking for the probability that a product is from plant C, given that it is defective. This is written as \( P(C|D) \).

The formula to calculate conditional probability using Bayes' Theorem is: \[ P(C|D) = \frac{P(D|C)P(C)}{P(D)} \] In other words, we need the probability that the customer received a defective product from plant C, \( P(D|C) \), and the overall probability of getting any product from plant C, \( P(C) \). These probabilities are used to find the probability of a product coming from plant C given that it is defective.

The Total Probability Theorem helps in finding the overall probability of an event by considering all possible ways that event can occur. It is useful when dealing with problems that involve a mixture of different sources or cases.

In our example, we need to find the total probability, \( P(D) \), of receiving a defective product. This involves considering the defective products from all three plants. The formula is: \[ P(D) = P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C) \]

By plugging in the values from the problem鈥攚here we know the probabilities of producing defective products from each plant and the probability of selecting a product from each plant鈥攚e get: \[ P(D) = (0.25 \times 0.35) + (0.05 \times 0.15) + (0.15 \times 0.50) = 0.17 \]

Defective products probability is essential in real-world scenarios like manufacturing and quality control. For our problem, we are looking for how likely it is that a received product came from a specific plant if the product is defective.

The provided probabilities were:

• \( P(D|A) = 0.25 \)
• \( P(D|B) = 0.05 \)
• \( P(D|C) = 0.15 \)

This means if we select a product randomly from plant A, the likelihood it is defective is 25%, from plant B is 5%, and from plant C is 15%. These probabilities indicate the quality control levels at each plant, and we used these values to calculate the total probability of any defective product.

Probability distribution refers to how probabilities are assigned to different outcomes. In this problem, our outcomes are determined by the production rates of three plants and the probability of receiving defective products from each.

We are given:

• \( P(A) = 0.35 \)
• \( P(B) = 0.15 \)
• \( P(C) = 0.50 \)

These probabilities reflect the portion of total products each plant contributes. Probability distribution is vital for solving such problems because it helps calculate overall probabilities and understand how defects are spread across different sources.

By grasping how these probabilities and distributions interact, we can apply Bayes' Theorem effectively to find the probability that a defective product came from plant C.