Quality control: A population of 591 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification, with the results shown in the following table. A wafer is chosen at random from the population. Write your answer as a fraction or a decimal, rounded to four decimal places.
Lot Conforming Nonconforming
A 85 17
B 167 30
C 260 32
(a) What is the probability that the wafer is from Lot C?
(b) What is the probability that the wafer is nonconforming?
(c) What is the probability that the wafer is from Lot C and is nonconforming?
(d) Given that the wafer is from Lot C, what is the probability that it is nonconforming?
(e) Given that the wafer is nonconforming, what is the probability that it is from Lot C?
(f) Let [tex]$E_1$[/tex] be the event that the wafer comes from Lot C, and let [tex]$E_2$[/tex] be the event that the wafer is nonconforming. Are [tex]$E_1$[/tex] and [tex]$E_2$[/tex] independent?
Answer (2)
Calculate the probability of a wafer being from Lot C: 591 292 ≈ 0.4941 .
Calculate the probability of a wafer being nonconforming: 591 79 ≈ 0.1337 .
Calculate the probability of a wafer being from Lot C and nonconforming: 591 32 ≈ 0.0541 .
Determine that events 'wafer from Lot C' and 'wafer is nonconforming' are not independent. Not independent
Explanation
Analyze the problem and data We are given a population of 591 semiconductor wafers categorized by lot (A, B, C) and conformance to a thickness specification (Conforming, Nonconforming). We need to calculate several probabilities and determine if the events 'wafer from Lot C' and 'wafer is nonconforming' are independent. Let's start by summarizing the data:
Total wafers = 591 Wafers from Lot A: 85 (Conforming) + 17 (Nonconforming) = 102 Wafers from Lot B: 167 (Conforming) + 30 (Nonconforming) = 197 Wafers from Lot C: 260 (Conforming) + 32 (Nonconforming) = 292 Conforming wafers: 85 (Lot A) + 167 (Lot B) + 260 (Lot C) = 512 Nonconforming wafers: 17 (Lot A) + 30 (Lot B) + 32 (Lot C) = 79
Calculate P(C) (a) We want to find the probability that a randomly chosen wafer is from Lot C. This is given by: P ( C ) = Total number of wafers Number of wafers from Lot C = 591 292 ≈ 0.4941 So, the probability that the wafer is from Lot C is approximately 0.4941.
Calculate P(Nonconforming) (b) Next, we want to find the probability that a randomly chosen wafer is nonconforming. This is given by: P ( Nonconforming ) = Total number of wafers Number of nonconforming wafers = 591 79 ≈ 0.1337 Thus, the probability that the wafer is nonconforming is approximately 0.1337.
Calculate P(C and Nonconforming) (c) We want to find the probability that a randomly chosen wafer is from Lot C and is nonconforming. This is given by: P ( C ∩ Nonconforming ) = Total number of wafers Number of nonconforming wafers from Lot C = 591 32 ≈ 0.0541 Therefore, the probability that the wafer is from Lot C and is nonconforming is approximately 0.0541.
Calculate P(Nonconforming | C) (d) We want to find the probability that a wafer is nonconforming given that it is from Lot C. This is a conditional probability: P ( Nonconforming ∣ C ) = P ( C ) P ( C ∩ Nonconforming ) = 292/591 32/591 = 292 32 ≈ 0.1096 So, given that the wafer is from Lot C, the probability that it is nonconforming is approximately 0.1096.
Calculate P(C | Nonconforming) (e) We want to find the probability that a wafer is from Lot C given that it is nonconforming. This is another conditional probability: P ( C ∣ Nonconforming ) = P ( Nonconforming ) P ( C ∩ Nonconforming ) = 79/591 32/591 = 79 32 ≈ 0.4051 Thus, given that the wafer is nonconforming, the probability that it is from Lot C is approximately 0.4051.
Check for independence (f) To determine if events E 1 (wafer from Lot C) and E 2 (wafer is nonconforming) are independent, we need to check if P ( E 1 ∩ E 2 ) = P ( E 1 ) ⋅ P ( E 2 ) . We have:
P ( E 1 ∩ E 2 ) = P ( C ∩ Nonconforming ) = 591 32 ≈ 0.0541
P ( E 1 ) ⋅ P ( E 2 ) = P ( C ) ⋅ P ( Nonconforming ) = 591 292 ⋅ 591 79 = 349281 23068 ≈ 0.0660
Since 0.0541 = 0.0660 , the events E 1 and E 2 are not independent.
Examples
In manufacturing, understanding the probabilities of defects arising from different production lines (lots) is crucial for quality control. For instance, if a particular lot (Lot C in this case) has a higher probability of producing nonconforming wafers, resources can be allocated to investigate and rectify the issues specific to that lot. This could involve recalibrating machinery, improving material sourcing, or enhancing worker training. By calculating conditional probabilities, manufacturers can also assess the likelihood of a wafer being from a specific lot given that it is nonconforming, which helps in tracing the source of defects and implementing targeted corrective actions. This ensures a more efficient and effective quality control process, reducing waste and improving overall product reliability.
The probabilities for the semiconductor wafers are as follows: Lot C probability is approximately 0.4941, nonconforming probability is about 0.1337, and nonconforming Lot C probability is around 0.0541. Given the wafer is from Lot C, the nonconforming probability is approximately 0.1096, while given a wafer is nonconforming, the Lot C probability is about 0.4051. The events are not independent as their probabilities do not satisfy the independence condition.
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