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In the realm of probability, the Total Probability Theorem is a handy tool that allows us to find the probability of an event based on the probabilities of a set of mutually exclusive and exhaustive sub-events. Simply put, it helps us calculate the overall probability of an event by considering all possible ways in which it can occur. In our scenario, we want to determine the probability of an incorrect fill, regardless of speed. This is achieved by considering incorrect fills during both high-speed and low-speed operations.
To apply the Total Probability Theorem, we need:

• The probability of an incorrect fill given a high-speed operation, denoted as \(P(E|H) = 0.01\).
• The probability of an incorrect fill given a low-speed operation, denoted as \(P(E|L) = 0.001\).
• The probabilities of high-speed \(P(H) = 0.3\) and low-speed \(P(L) = 0.7\) operations.

Using these, the Total Probability Theorem can be expressed as: \[ P(E) = P(E|H) \cdot P(H) + P(E|L) \cdot P(L) \] Filling in the given values, we calculate \( P(E) = 0.01 \times 0.3 + 0.001 \times 0.7 = 0.0037 \). This tells us that the overall probability of an incorrect fill is \(0.0037\). This foundation sets us up for understanding conditional probability in the next section.

Bayes' Theorem is like a detective in the world of probability. It helps us reverse a conditional probability to find what's originally hidden. In our case, we want to discover the probability that an incorrect fill was created during a high-speed operation. For this task, Bayes' Theorem steps in cheerfully.
To apply Bayes' Theorem, we need to know:

• The initial probabilities: incorrect fills at high speed \( P(E|H) = 0.01 \) and the overall probability of an incorrect fill \( P(E) = 0.0037 \).
• The probability of high-speed operation \( P(H) = 0.3 \).

Bayes' formula unfolds as: \[ P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)} \] By substituting our numbers into Bayes' magical formula, we compute: \[ P(H|E) = \frac{0.01 \times 0.3}{0.0037} \approx 0.8108 \] So, if an incorrect fill is discovered, there's an 81.1% chance it was during high-speed processing. Bayes provides a look back in time, revealing the path behind observed outcomes.

Conditional probability is the art of finding probability in the light of new information. It refers to the likelihood of an event occurring, given that another event has already occurred. Imagine you discover an incorrectly filled container. Given this new piece of information, what's its origin based on speed? Conditional probability supplies the answer.
Using conditional probability, we already computed using Bayes' theorem:

• \(P(H|E)\): the probability of high-speed fill, given an error occurred, is about \(0.811\) or \(81.1\%\).
• From earlier data, we also know \(P(L|E)\): the probability of low-speed fill given error occurred, completes the picture totaling to 100% with \(\approx 18.9\%\).

These computations stem from the core nature of conditional probability, giving us insights as conditions change. In statistical analysis, this concept answers questions contingent on evolving scenarios. Such logic lets us dynamically adjust perceptions based on newfound evidence, bridging raw data with real-world relevance.