Quality control: A population of 603 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.

Question

Lot A- 87 conforming; 10 nonconforming
Lot B- 165 conforming; 40 non
Lot C- 258 conforming; 43 non

(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 Ev1 be the event that the wafer comes from Lot C, and let Ev2 be the event that the wafer is nonconforming. Are Ev1 and Ev2 independent?

Answer 1

To solve the problems, we first summarize the data provided in the problem:

• Lot A: 87 conforming, 10 nonconforming (Total = 97)
• Lot B: 165 conforming, 40 nonconforming (Total = 205)
• Lot C: 258 conforming, 43 nonconforming (Total = 301)

Total number of wafers = 603 (which matches the total of the three lots).

Let's calculate the required probabilities.

Part (a)

What is the probability that the wafer is from Lot C?

The number of wafers from Lot C is 301. \[ P(\text{Lot C}) = \frac{\text{Number of wafers from Lot C}}{\text{Total number of wafers}} = \frac{301}{603} \approx 0.4992 \]

Part (b)

What is the probability that the wafer is nonconforming?

To find this, we need to calculate the total number of nonconforming wafers.

Nonconforming wafers:

• Lot A: 10
• Lot B: 40
• Lot C: 43
• Total nonconforming = 10 + 40 + 43 = 93

\[ P(\text{nonconforming}) = \frac{\text{Total nonconforming}}{\text{Total number of wafers}} = \frac{93}{603} \approx 0.1542 \]

Part (c)

What is the probability that the wafer is from Lot C and is nonconforming?

The number of nonconforming wafers from Lot C is 43.

\[ P(\text{Lot C and nonconforming}) = \frac{\text{Nonconforming from Lot C}}{\text{Total number of wafers}} = \frac{43}{603} \approx 0.0713 \]

Part (d)

Given that the wafer is from Lot C, what is the probability that it is nonconforming?

Given that the wafer is from Lot C, we only consider the total wafers in Lot C, which is 301.

\[ P(\text{nonconforming} | \text{Lot C}) = \frac{\text{Nonconforming from Lot C}}{\text{Total wafers from Lot C}} = \frac{43}{301} \approx 0.1432 \]

Part (e)

Given that the wafer is nonconforming, what is the probability that it is from Lot C?

Now we need to find the probability that a nonconforming wafer comes from Lot C.

Using the previous counts:

• Total number of nonconforming wafers = 93.
• Nonconforming wafers from Lot C = 43.

\[ P(\text{Lot C} | \text{nonconforming}) = \frac{\text{Nonconforming from Lot C}}{\text{Total nonconforming}} = \frac{43}{93} \approx 0.4624 \]

Part (f)

Are Ev1 and Ev2 independent?

Two events \(Ev1\) (Lot C) and \(Ev2\) (nonconforming) are independent if: \[ P(Ev1 \cap Ev2) = P(Ev1) \times P(Ev2) \]

We found:

• \(P(\text{Lot C}) \approx 0.4992\)
• \(P(\text{nonconforming}) \approx 0.1542\)
• \(P(\text{Lot C and nonconforming}) \approx 0.0713\)

Calculating \(P(\text{Lot C}) \times P(\text{nonconforming})\): \[ P(\text{Lot C}) \times P(\text{nonconforming}) \approx 0.4992 \times 0.1542 \approx 0.0769 \]

Now we compare it to \(P(\text{Lot C and nonconforming})\):

• \(P(\text{Lot C and nonconforming}) \approx 0.0713\)

Since \(P(\text{Lot C and nonconforming}) \neq P(\text{Lot C}) \times P(\text{nonconforming})\), the events are not independent.

Summary:

(a) \( \approx 0.4992 \)
(b) \( \approx 0.1542 \)
(c) \( \approx 0.0713 \)
(d) \( \approx 0.1432 \)
(e) \( \approx 0.4624 \)
(f) Not independent.