Question

Suppose our manufacturing company purchases a certain part from three different suppliers \(S_{1}, S_{2},\) and \(S_{3}\). Supplier \(S_{1}\) provides \(40 \%\) of our parts, and suppliers \(S_{2}\) and \(S_{3}\) provide \(35 \%\) and \(25 \%,\) respectively. Furthermore, \(20 \%\) of the parts shipped by \(S_{1}\) are defective, \(10 \%\) of the parts shipped by \(S_{2}\) are defective, and \(5 \%\) of the parts from \(S_{3}\) are defective. Now, suppose an employee at our company chooses a part at random. (a) What is the probability that the part is good? (b) If the part is good, what is the probability that it was shipped by \(S_{1} ?\) (c) If the part is defective, what is the probability that it was shipped by \(S_{1} ?\)

Short answer

(a) 0.8725; (b) 0.3667; (c) 0.6275

Step-by-step solution

Determine the Probability of a Good Part from Each Supplier

To find the probability that a part is good from each supplier, calculate one minus the probability of a defective part from that supplier.\For Supplier \(S_1\), the probability of a good part is \(1 - 0.2 = 0.8\).\For Supplier \(S_2\), the probability of a good part is \(1 - 0.1 = 0.9\).\For Supplier \(S_3\), the probability of a good part is \(1 - 0.05 = 0.95\).

Calculate the Total Probability of a Good Part

Using the law of total probability, calculate the total probability that a part is good.\The probability is given by:\[ \text{P(Good Part)} = 0.4 \times 0.8 + 0.35 \times 0.9 + 0.25 \times 0.95 \]Calculate:\[0.4 \times 0.8 = 0.32\]\[0.35 \times 0.9 = 0.315\]\[0.25 \times 0.95 = 0.2375\] \Add these values:\[ \text{P(Good Part)} = 0.32 + 0.315 + 0.2375 = 0.8725\]

Find Probability Part is Shipped by S1 Given it's Good

Use Bayes' theorem to find the probability that the part was shipped by \(S_1\) given it is good.\\[ \text{P}(S_1|\text{Good Part}) = \frac{\text{P(Good Part | } S_1) \times \text{P}(S_1)}{\text{P(Good Part)}} \]Substitute the values:\[ \text{P}(S_1|\text{Good Part}) = \frac{0.8 \times 0.4}{0.8725} \]Calculating:\[ \text{P}(S_1|\text{Good Part}) = \frac{0.32}{0.8725} \approx 0.3667\]

Calculate Probability Part is Defective from Each Supplier

First, find the probability that a part is defective from each supplier.\Supplier \(S_1\): \(0.4 \times 0.2 = 0.08\)\Supplier \(S_2\): \(0.35 \times 0.1 = 0.035\)\Supplier \(S_3\): \(0.25 \times 0.05 = 0.0125\).

Compute Total Probability of a Defective Part

Sum the probabilities from Step 4 to get the total probability that a part is defective.\\[ \text{P(Defective Part)} = 0.08 + 0.035 + 0.0125 = 0.1275\]

Find Probability Part is Shipped by S1 Given it's Defective

Apply Bayes' theorem to calculate the probability that a defective part was shipped by \(S_1\).\\[ \text{P}(S_1|\text{Defective Part}) = \frac{\text{P(Defective Part | } S_1) \times \text{P}(S_1)}{\text{P(Defective Part)}} \]Substitute the values:\[ \text{P}(S_1|\text{Defective Part}) = \frac{0.08}{0.1275} \]Calculating:\[ \text{P}(S_1|\text{Defective Part}) \approx 0.6275 \]

Key concepts

Law of Total Probability

The law of total probability is a powerful tool in probability theory. It helps in finding the probability of an event by considering all possible ways it can occur. In the context of manufacturing, where multiple suppliers provide parts, it can be used to calculate the overall probability of receiving a good part. Here's how this law comes into play:

• Imagine we have three suppliers (let's call them \(S_1\), \(S_2\), and \(S_3\)) with different shares in the total parts supplied: 40%, 35%, and 25% respectively.
• Each supplier has a different defect rate. Thus, the probability of a part being good from each supplier is determined by subtracting the defect rate from one. For example, the probability of getting a good part from \(S_1\) is \(0.8\) (because \(1 - 0.2 = 0.8\)).
• The law of total probability then combines these individual probabilities. It multiplies the probability of receiving a part from each supplier (e.g., 40% for \(S_1\)) by the probability that this part is good and then sums all these values.
• Mathematically, it is expressed as \(\text{P(Good Part)} = 0.4 \times 0.8 + 0.35 \times 0.9 + 0.25 \times 0.95\). Calculating this gives us an overall probability of approximately 87.25% that a part is good when selected at random.

This approach is crucial in many sectors, helping decision-makers understand the quality dynamics when sourcing from multiple suppliers.

Bayes' Theorem

Bayes' Theorem is a fundamental theorem providing a way to reverse the conditions between two probabilities. It allows us to update our belief about the probability of an event given new evidence. In the scenario of choosing parts from different suppliers, Bayes' Theorem is especially useful. It helps calculate the probability of a part coming from a specific supplier, once we know whether the part is good or defective.

• After calculating the total probability of the part being good using the law of total probability, Bayes' Theorem lets us find out how likely it is that a part comes from \(S_1\) given that it's a good part.
• The theorem applies the formula: \[ \text{P}(S_1|\text{Good Part}) = \frac{\text{P(Good Part | } S_1) \times \text{P}(S_1)}{\text{P(Good Part)}} \]
• This means we compute the probability that a good part is from \(S_1\), multiplied by the chance of getting a part from \(S_1\), then divided by the total probability of a good part.
• The calculated probability is approximately 36.67%, indicating how knowing a part is good slightly changes our understanding of its source.

Bayes' Theorem thus provides a robust method for refining probabilities based on new or additional information.

Defective Products Analysis

In quality control, understanding the likelihood of defective products is crucial for maintaining standards. By analyzing defect rates from different suppliers, a company can make informed decisions to minimize defects and improve overall quality.

• First, you determine the defect rate for each supplier, which reflects the percentage of parts that are defective. For example, the defect rate for \(S_1\) is 20%, meaning that 20% of the parts from \(S_1\) are defective.
• Next, calculate the probability of a part being defective using the suppliers' distribution percentages. This analysis involves multiplying each supplier's defect probability by its proportional supply contribution and summing them up.
• This gives the overall probability of receiving a defective part when picking a random part. In our scenario with suppliers \(S_1\), \(S_2\), and \(S_3\), this probability is approximately 12.75%.
• To understand the impact of a defect on sourcing, use Bayes' Theorem again. For example, assessing the probability that a defective part came from \(S_1\) would involve considering the likelihood of selection from \(S_1\), even given a defect.
• The resulting probability is about 62.75%, indicating a significant portion of defects might come from \(S_1\). Knowing this can guide corrective actions or changes in supplier management.

Defective products analysis, combined with probability theories, helps companies enhance quality assurance and optimize their supply chain decisions.