Quality Control: A population of 596 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 | 91 | 13 |
| B | 164 | 43 |
| C | 252 | 33 |

(a) What is the probability that the wafer is from Lot A?
(b) What is the probability that the wafer is conforming?
(c) What is the probability that the wafer is from Lot A and is conforming?
(d) Given that the wafer is from Lot A, what is the probability that it is conforming?
(e) Given that the wafer is conforming, what is the probability that it is from Lot A?
(f) Let [tex]\( E_1 \)[/tex] be the event that the wafer comes from Lot A, and let [tex]\( E_2 \)[/tex] be the event that the wafer is conforming. Are [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] independent?



Answer :

Sure, let's go through the solution to each part of the question step-by-step.

### Given Data:
- Total wafers = 596
- Lot A:
- Conforming = 91
- Nonconforming = 13
- Lot B:
- Conforming = 164
- Nonconforming = 43
- Lot C:
- Conforming = 252
- Nonconforming = 33

### Calculations:

(a) What is the probability that the wafer is from Lot A?

The probability of choosing a wafer from Lot A is calculated by dividing the total number of wafers in Lot A by the total number of wafers:
[tex]\[ \text{Probability(Lot A)} = \frac{\text{Total Lot A}}{\text{Total Wafers}} = \frac{91 + 13}{596} = \frac{104}{596} \approx 0.1745 \][/tex]

(b) What is the probability that the wafer is conforming?

To find the probability that a wafer is conforming, sum the number of conforming wafers from all lots and divide by the total number of wafers:
[tex]\[ \text{Probability(Conforming)} = \frac{\text{Conforming Lot A} + \text{Conforming Lot B} + \text{Conforming Lot C}}{\text{Total Wafers}} = \frac{91 + 164 + 252}{596} = \frac{507}{596} \approx 0.8507 \][/tex]

(c) What is the probability that the wafer is from Lot A and is conforming?

The probability that a wafer is both from Lot A and conforming is:
[tex]\[ \text{Probability(Lot A and Conforming)} = \frac{\text{Conforming Lot A}}{\text{Total Wafers}} = \frac{91}{596} \approx 0.1527 \][/tex]

(d) Given that the wafer is from Lot A, what is the probability that it is conforming?

This is a conditional probability and is calculated as follows:
[tex]\[ \text{Probability(Conforming | Lot A)} = \frac{\text{Conforming Lot A}}{\text{Total Lot A}} = \frac{91}{91 + 13} = \frac{91}{104} \approx 0.8750 \][/tex]

(e) Given that the wafer is conforming, what is the probability that it is from Lot A?

Again, this is a conditional probability:
[tex]\[ \text{Probability(Lot A | Conforming)} = \frac{\text{Conforming Lot A}}{\text{Total Conforming}} = \frac{91}{507} \approx 0.1795 \][/tex]

(f) Let [tex]\( E_1 \)[/tex] be the event that the wafer comes from Lot A, and let [tex]\( E_2 \)[/tex] be the event that the wafer is conforming. Are [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] independent?

Two events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] are independent if:
[tex]\[ P(E_1 \cap E_2) = P(E_1) \times P(E_2) \][/tex]

Let's check this:
[tex]\[ P(E_1 \cap E_2) = 0.1527 \][/tex]
[tex]\[ P(E_1) \times P(E_2) = 0.1745 \times 0.8507 \approx 0.1484 \][/tex]

Since [tex]\( P(E_1 \cap E_2) \neq P(E_1) \times P(E_2) \)[/tex], the events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] are not independent.

### Summary of Results:
(a) Probability that the wafer is from Lot A: [tex]\( \approx 0.1745 \)[/tex]
(b) Probability that the wafer is conforming: [tex]\( \approx 0.8507 \)[/tex]
(c) Probability that the wafer is from Lot A and is conforming: [tex]\( \approx 0.1527 \)[/tex]
(d) Given that the wafer is from Lot A, probability that it is conforming: [tex]\( \approx 0.8750 \)[/tex]
(e) Given that the wafer is conforming, probability that it is from Lot A: [tex]\( \approx 0.1795 \)[/tex]
(f) Are events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] independent? No, they are not independent.

I hope this detailed solution helps you understand the problem and its resolution!