Problem 211 - Applied Statistics and Probability for Engineers

Question

In an automated filling operation, the probability of an incorrect fill when the process is operated at a low speed is \(0.001 .\) When the process is operated at a high speed, the probability of an incorrect fill is 0.01 . Assume that \(30 \%\) of the containers are filled when the process is operated at a high speed and the remainder are filled when the process is operated at a low speed. (a) What is the probability of an incorrectly filled container? (b) If an incorrectly filled container is found, what is the probability that it was filled during the high-speed operation?

Step-by-Step Solution

Answer

(a) Probability of incorrect fill is 0.0037. (b) Probability it was filled at high speed is approximately 0.8108.

Step 1: Identify Events and Probabilities

Let event \(I\) represent an incorrect fill, event \(L\) represent low-speed operation, and event \(H\) represent high-speed operation. We know \(P(I|L) = 0.001\), \(P(I|H) = 0.01\), \(P(H) = 0.30\), and \(P(L) = 0.70\) because \(70\%\) of containers are filled at low speed.

Step 2: Use Total Probability Theorem for Part (a)

To find \(P(I)\), the probability of an incorrectly filled container, apply the Total Probability Theorem: \[ P(I) = P(I|L)P(L) + P(I|H)P(H) \] Substitute known values: \[ P(I) = (0.001 \times 0.70) + (0.01 \times 0.30) \]. Calculate to get \(P(I) = 0.0007 + 0.003 = 0.0037\).

Step 3: Use Bayes' Theorem for Part (b)

We need \(P(H|I)\), the probability the incorrect fill was made during high-speed operation. By Bayes' Theorem, \[ P(H|I) = \frac{P(I|H)P(H)}{P(I)} \]. Substitute known values: \[ P(H|I) = \frac{0.01 \times 0.30}{0.0037} \]. Calculate to find \(P(H|I) \approx 0.8108\).

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that allows us to update our beliefs based on new evidence. This theorem is particularly useful when you want to find the probability of an event, given that another event has already occurred. In simpler terms, it helps us determine the likelihood of a hypothesis given observed data.

In the context of the original exercise, Bayes' Theorem is applied to find the probability that an incorrectly filled container was filled during a high-speed operation, denoted as \( P(H|I) \). The formula we use is:

• \( P(H|I) = \frac{P(I|H) P(H)}{P(I)} \)

Here, \( P(I|H) \) is the probability of an incorrect fill given a high-speed operation, \( P(H) \) is the probability of a high-speed operation, and \( P(I) \) is the probability of an incorrect fill overall. By inserting the known probabilities into Bayes' formula, we can compute the probability of interest.

Understanding Bayes' Theorem helps in various fields including statistics, machine learning, and risk management, where making informed decisions based on incomplete data is crucial.

Total Probability Theorem

The Total Probability Theorem is another essential idea in probability, especially useful when dealing with complex systems where several mutually exclusive events can lead to the same outcome. It breaks down the probability of a desired event by considering all possible ways that event could occur.

In our original problem, the Total Probability Theorem is used to calculate the overall probability of an incorrectly filled container \( P(I) \). The formula applied is:

• \( P(I) = P(I|L)P(L) + P(I|H)P(H) \)

This equation accounts for each process speed, low \( L \) and high \( H \), where \( P(I|L) \) and \( P(I|H) \) are the probabilities of an incorrect fill given the respective speed, and \( P(L) \) and \( P(H) \) represent the probabilities of filling at those speeds.

In essence, the Total Probability Theorem allows us to combine these distinct paths into a single probability, helping in understanding how different factors contribute to a final outcome.

Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. It forms the backbone of concepts like Bayes' Theorem and is crucial for understanding dependencies between events.

In the exercise, conditional probability is expressed as \( P(I|L) \) and \( P(I|H) \), representing the probabilities of an incorrect fill given low or high-speed operation, respectively. The notation \( P(A|B) \) represents "the probability of A given B."

Understanding conditional probability helps you predict the likelihood of a specific outcome based on a set condition, which is vital in fields like predictive modeling and decision-making.

This concept ensures that we do not treat dependent events as if they were independent, which would lead to miscalculations and incorrect assumptions.