Problem 193 - Applied Statistics and Probability for Engineers

Question

In the manufacturing of a chemical adhesive, \(3 \%\) of all batches have raw materials from two different lots. This occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks. Only \(5 \%\) of batches with material from a single lot require reprocessing. However, the viscosity of batches consisting of two or more lots of material is more difficult to control, and \(40 \%\) of such batches require additional processing to achieve the required viscosity. Let \(A\) denote the event that a batch is formed from two dif- ferent lots, and let \(B\) denote the event that a lot requires additional processing. Determine the following probabilities: (a) \(P(A)\) (b) \(P\left(A^{\prime}\right)\) (c) \(P(B \mid A)\) (d) \(P\left(B \mid A^{\prime}\right)\) (e) \(P(A \cap B)\) (f) \(P\left(A \cap B^{\prime}\right)\) (g) \(P(B)\)

Step-by-Step Solution

Answer

(a) 0.03, (b) 0.97, (c) 0.40, (d) 0.05, (e) 0.012, (f) 0.018, (g) 0.0605

Step 1: Probability of Event A

Calculate the probability that a batch is from two different lots. Given is that only \(3\%\) of all batches are made of materials from two different lots. So, \(P(A) = 0.03\).

Step 2: Probability of Complement Event A'

The complement of event A, \(A'\), is when a batch is not from two different lots. Therefore, \(P(A') = 1 - P(A) = 1 - 0.03 = 0.97\).

Step 3: Probability of B given A

Calculate the probability that a batch requires reprocessing given that it is made from two different lots. This is \(P(B | A) = 0.40\).

Step 4: Probability of B given Complement A'

Calculate the probability that a batch requires reprocessing given that it is made from a single lot. This is \(P(B | A') = 0.05\).

Step 5: Probability of Event A intersection B

Using the definition of conditional probability, \(P(A \cap B) = P(B | A) \times P(A) = 0.40 \times 0.03 = 0.012\).

Step 6: Probability of Event A intersection B'

The complement of B, \(B'\), means a batch does not require reprocessing. So, \(P(A \cap B') = P(A) - P(A \cap B) = 0.03 - 0.012 = 0.018\).

Step 7: Total Probability of Event B

Calculate the total probability of B: \(P(B) = P(A \cap B) + P(A' \cap B)\). First, find \(P(A' \cap B) = P(B | A') \times P(A') = 0.05 \times 0.97 = 0.0485\). Then, \(P(B) = 0.012 + 0.0485 = 0.0605\).

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that allows us to find the probability of an event when we know that another event has already occurred. This is particularly useful when dealing with events that are connected or dependent on each other.

For instance, when working with events from a manufacturing process, like the production of batches with materials from different lots, we use conditional probability to assess what happens given that particular conditions are met.

In our case scenario, we have two events:

• Event A: A batch is formed from two different lots.
• Event B: A batch requires additional processing.

The probability of event B occurring given that event A has already occurred is expressed as the conditional probability, denoted as \(P(B\mid A)\). Here, it indicates the likelihood of needing reprocessing if a batch is made from two different lots. Knowing this helps manufacturers predict and minimize inefficiencies. In this example, \(P(B\mid A) = 0.40\), meaning there's a 40% chance of needing additional processing under these conditions.

Complementary Events

In probability theory, complementary events are closely related to their corresponding primary events. For any given event, the complementary event accounts for all outcomes not included in the event itself.

For instance, let event \(A\) be when a batch is formed from two different lots, with a probability \(P(A) = 0.03\). The complementary event, \(A'\), reflects when batches are not formed from two different lots and is calculated as: \[P(A') = 1 - P(A) = 0.97\]

This means 97% of batches come from a single lot.

Complementary events are useful because they provide a full picture by considering both what occurs and what does not occur in a scenario. For manufacturers, understanding \(A'\) helps highlight the normal conditions where most batches don't require complicated handling. As for the probability of reprocessing when the batch is from a single lot, \(P(B \mid A') = 0.05\), it shows that additional efforts are far less likely needed in these circumstances.

Intersection of Events

The intersection of events focuses on the simultaneous occurrence of two or more events. In probability terms, the intersection is noted as \(A \cap B\), representing a scenario where both events A and B happen together.

Calculating the probability of two events occurring together is essential for understanding complex systems, like manufacturing processes. Here, the probability of a batch being both from two lots and requiring reprocessing is calculated as follows:

• First, use conditional probability: \(P(A \cap B) = P(B \mid A) \times P(A)\)
• Plug in the values: \(P(A \cap B) = 0.40 \times 0.03 = 0.012\)

Thus, there's a 1.2% probability that a batch will be composed of materials from two lots and needs additional processing.

This insight aids manufacturers in pinpointing critical operational snegatives and determining which scenarios significantly impact production efficiency.

Manufacturing Process

The manufacturing process involves the use of materials, machines, and labor to produce goods consistently. In the chemical adhesive production described, attention is needed on how batches are composed and processed.

A pivotal aspect of manufacturing is materials management, ensuring that raw materials are optimally utilized. Sometimes, batches involve complex scenarios like replenishing tanks from different lots, and this poses a challenge in maintaining quality standards, as seen from higher rates of reprocessing.

In this scenario:

• Only 3% of batches use materials from two different lots.
• These batches have a 40% likelihood of reprocessing due to viscosity issues.
• Batches formed from a single lot face far fewer challenges, with only a 5% chance of reprocessing.

Understanding these aspects enables manufacturers to improve the production system to minimize waste and effort. Efficiently managing complementary events and intersections allows them to anticipate potential bottlenecks and improve overall operational flow.