A manufacturer of tablets receives its LED screens from three different suppliers:
A
A,
B
B, and
C
C. The probability that any one LED screen received by the plant comes from these three suppliers are 0.60, 0.30, and 0.10 respectively. Suppose that 96% of the LED screens from
A
A, 80% of those from
B
B, and 65% of those from
C
C perform according to specifications.

(a) What is the probability that the LED screen conforms to specifications?

(b) If a randomly chosen LED screen conforms to specifications, what is the probability that it came from
i
i Supplier
A
A,
i
i
ii Supplier
B
B, and
i
i
i
iii Supplier
C
C?

Let's solve this problem step-by-step.

### Part (a): Probability that the LED screen conforms to specifications

We can use the Law of Total Probability to find the probability that an LED screen conforms to specifications. Let's denote the events as follows:
- \(A\): The screen is from supplier \(A\).
- \(B\): The screen is from supplier \(B\).
- \(C\): The screen is from supplier \(C\).
- \(S\): The screen conforms to specifications.

Given:
- \(P(A) = 0.60\)
- \(P(B) = 0.30\)
- \(P(C) = 0.10\)
- \(P(S|A) = 0.96\)
- \(P(S|B) = 0.80\)
- \(P(S|C) = 0.65\)

The probability that the screen conforms to specifications is:
\[P(S) = P(S|A)P(A) + P(S|B)P(B) + P(S|C)P(C)\]

Substitute the given values:
\[P(S) = (0.96 \times 0.60) + (0.80 \times 0.30) + (0.65 \times 0.10)\]

Calculate each term:
\[0.96 \times 0.60 = 0.576\]
\[0.80 \times 0.30 = 0.24\]
\[0.65 \times 0.10 = 0.065\]

Sum these probabilities:
\[P(S) = 0.576 + 0.24 + 0.065 = 0.881\]

So, the probability that the LED screen conforms to specifications is \(0.881\) or 88.1%.

### Part (b): Conditional probability of the screen coming from each supplier given that it conforms to specifications

We use Bayes' Theorem to find these probabilities.

#### (i) Probability that it came from Supplier \(A\):

\[P(A|S) = \frac{P(S|A)P(A)}{P(S)}\]

Substitute the given values:
\[P(A|S) = \frac{0.96 \times 0.60}{0.881} = \frac{0.576}{0.881} \approx 0.6549\]

#### (ii) Probability that it came from Supplier \(B\):

\[P(B|S) = \frac{P(S|B)P(B)}{P(S)}\]

Substitute the given values:
\[P(B|S) = \frac{0.80 \times 0.30}{0.881} = \frac{0.24}{0.881} \approx 0.2724\]

#### (iii) Probability that it came from Supplier \(C\):

\[P(C|S) = \frac{P(S|C)P(C)}{P(S)}\]

Substitute the given values:
\[P(C|S) = \frac{0.65 \times 0.10}{0.881} = \frac{0.065}{0.881} \approx 0.0738\]

### Summary of the results:
- (a) The probability that the LED screen conforms to specifications is 0.881.
- (b) Given that an LED screen conforms to specifications, the probabilities that it came from:
  - Supplier \(A\) is approximately 0.6549.
  - Supplier \(B\) is approximately 0.2724.
  - Supplier \(C\) is approximately 0.0738.

Would you like any further details or have any questions?

### Related Questions:
1. How does the overall quality of screens change if Supplier \(A\)'s performance drops to 90%?
2. What is the impact on the probability of conformance if the proportions of screens from each supplier change?
3. How does the probability of a screen coming from Supplier \(A\) change if Supplier \(B\) improves its performance to 85%?
4. What is the probability that a screen from Supplier \(C\) does not conform to specifications?
5. How would you calculate the probability of a screen coming from Supplier \(A\) if it does not conform to specifications?
6. If the total number of screens received is 1000, how many screens from each supplier would be expected to conform to specifications?
7. How does increasing the proportion of screens from Supplier \(C\) affect the overall conformance rate?
8. What is the probability that a randomly chosen screen comes from Supplier \(B\) and conforms to specifications?

### Tip:
When dealing with probabilities, breaking the problem down into smaller parts and using the laws of probability systematically can simplify complex scenarios.

Calculate the probability that an LED screen conforms to specifications based on probabilities from three suppliers (A, B, C) and their respective performance rates. Understand the conditional probabilities of a screen coming from each supplier given that it conforms to specifications. Learn how to apply the Law of Total Probability and Bayes' Theorem in probability calculations.

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