Bot GPT 4-o mini Let's first summarize the data presented in the problem.
Summary of Wafers:
| Lot | Conforming | Nonconforming | Total | |------|------------|---------------|--------| | A | 80 | 13 | 93 | | B | 170 | 31 | 201 | | C | 265 | 41 | 306 | | Total | 515 | 85 | 600 |
Calculations:
(a) What is the probability that the wafer is from Lot C?
The total number of wafers from Lot C is 306.
\[ P(\text{Lot C}) = \frac{\text{Number of wafers in Lot C}}{\text{Total number of wafers}} = \frac{306}{600} = 0.51 \]
Rounded to four decimal places:
\[ P(\text{Lot C}) = 0.5100 \]
(b) What is the probability that the wafer is nonconforming?
The total number of nonconforming wafers is 85.
\[ P(\text{Nonconforming}) = \frac{\text{Number of nonconforming wafers}}{\text{Total number of wafers}} = \frac{85}{600} \approx 0.1417 \]
Rounded to four decimal places:
\[ P(\text{Nonconforming}) = 0.1417 \]
(c) What is the probability that the wafer is from Lot C and is nonconforming?
The number of nonconforming wafers from Lot C is 41.
\[ P(\text{Lot C and Nonconforming}) = \frac{\text{Number of nonconforming wafers in Lot C}}{\text{Total number of wafers}} = \frac{41}{600} \approx 0.0683 \]
Rounded to four decimal places:
\[ P(\text{Lot C and Nonconforming}) = 0.0683 \]
(d) Given that the wafer is from Lot C, what is the probability that it is nonconforming?
Using conditional probability,
\[ P(\text{Nonconforming | Lot C}) = \frac{P(\text{Lot C and Nonconforming})}{P(\text{Lot C})} = \frac{41/600}{306/600} = \frac{41}{306} \approx 0.1346 \]
Rounded to four decimal places:
\[ P(\text{Nonconforming | Lot C}) = 0.1346 \]
(e) Given that the wafer is nonconforming, what is the probability that it is from Lot C?
Again using conditional probability,
\[ P(\text{Lot C | Nonconforming}) = \frac{P(\text{Lot C and Nonconforming})}{P(\text{Nonconforming})} = \frac{41/600}{85/600} = \frac{41}{85} \approx 0.4824 \]
Rounded to four decimal places:
\[ P(\text{Lot C | Nonconforming}) = 0.4824 \]
(f) Are Ev1 (wafer comes from Lot C) and Ev2 (wafer is nonconforming) independent?
To check for independence, we can see if
\[ P(\text{Lot C and Nonconforming}) = P(\text{Lot C}) \cdot P(\text{Nonconforming}) \]
Calculating:
\[ P(\text{Lot C}) \cdot P(\text{Nonconforming}) = 0.51 \times 0.1417 \approx 0.0723 \]
Since \( P(\text{Lot C and Nonconforming}) \approx 0.0683 \) is not equal to \( P(\text{Lot C}) \cdot P(\text{Nonconforming}) \approx 0.0723 \), the events are not independent.
Thus, we conclude that:
Ev1 and Ev2 are not independent.
1 answer