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!
### 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!
