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Sagot :
Let's solve the question step by step.
First, we need to determine the probability that a lens randomly selected is defective. We'll use the provided information about the proportion of lenses each company produces and the probability that a lens from each company is defective.
The probabilities are as follows:
- Probability that a lens is made by Greens: [tex]\( P(\text{Greens}) = 0.60 \)[/tex]
- Probability that a lens is made by Parsons: [tex]\( P(\text{Parsons}) = 0.15 \)[/tex]
- Probability that a lens is made by Ratten: [tex]\( P(\text{Ratten}) = 0.25 \)[/tex]
The probabilities that a lens made by each company is defective are:
- Probability that a lens made by Greens is defective: [tex]\( P(\text{Def} | \text{Greens}) = 0.05 \)[/tex]
- Probability that a lens made by Parsons is defective: [tex]\( P(\text{Def} | \text{Parsons}) = 0.10 \)[/tex]
- Probability that a lens made by Ratten is defective: [tex]\( P(\text{Def} | \text{Ratten}) = 0.06 \)[/tex]
Next, we calculate the total probability that a lens is defective, [tex]\( P(\text{Def}) \)[/tex]. This can be found using the law of total probability:
[tex]\[ P(\text{Def}) = P(\text{Greens}) \cdot P(\text{Def} | \text{Greens}) + P(\text{Parsons}) \cdot P(\text{Def} | \text{Parsons}) + P(\text{Ratten}) \cdot P(\text{Def} | \text{Ratten}) \][/tex]
Substituting in the values:
[tex]\[ P(\text{Def}) = 0.60 \cdot 0.05 + 0.15 \cdot 0.10 + 0.25 \cdot 0.06 \][/tex]
[tex]\[ P(\text{Def}) = 0.03 + 0.015 + 0.015 \][/tex]
[tex]\[ P(\text{Def}) = 0.06 \][/tex]
Now, we need to find the conditional probability that a defective lens was made by Greens, [tex]\( P(\text{Greens} | \text{Def}) \)[/tex]. To do this, we'll use Bayes' theorem:
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{P(\text{Def} | \text{Greens}) \cdot P(\text{Greens})}{P(\text{Def})} \][/tex]
Substituting in the values:
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{0.05 \cdot 0.60}{0.06} \][/tex]
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{0.03}{0.06} \][/tex]
[tex]\[ P(\text{Greens} | \text{Def}) = 0.50 \][/tex]
Now we need to compare this result with the given choices to determine the correct answer. The choices are:
A. 0.24
B. 0.60
C. 0.71
D. 0.83
The closest value to our calculated probability, 0.50, in the given options is:
B. 0.60
Therefore, the correct answer is:
[tex]\( \boxed{2} \)[/tex]
First, we need to determine the probability that a lens randomly selected is defective. We'll use the provided information about the proportion of lenses each company produces and the probability that a lens from each company is defective.
The probabilities are as follows:
- Probability that a lens is made by Greens: [tex]\( P(\text{Greens}) = 0.60 \)[/tex]
- Probability that a lens is made by Parsons: [tex]\( P(\text{Parsons}) = 0.15 \)[/tex]
- Probability that a lens is made by Ratten: [tex]\( P(\text{Ratten}) = 0.25 \)[/tex]
The probabilities that a lens made by each company is defective are:
- Probability that a lens made by Greens is defective: [tex]\( P(\text{Def} | \text{Greens}) = 0.05 \)[/tex]
- Probability that a lens made by Parsons is defective: [tex]\( P(\text{Def} | \text{Parsons}) = 0.10 \)[/tex]
- Probability that a lens made by Ratten is defective: [tex]\( P(\text{Def} | \text{Ratten}) = 0.06 \)[/tex]
Next, we calculate the total probability that a lens is defective, [tex]\( P(\text{Def}) \)[/tex]. This can be found using the law of total probability:
[tex]\[ P(\text{Def}) = P(\text{Greens}) \cdot P(\text{Def} | \text{Greens}) + P(\text{Parsons}) \cdot P(\text{Def} | \text{Parsons}) + P(\text{Ratten}) \cdot P(\text{Def} | \text{Ratten}) \][/tex]
Substituting in the values:
[tex]\[ P(\text{Def}) = 0.60 \cdot 0.05 + 0.15 \cdot 0.10 + 0.25 \cdot 0.06 \][/tex]
[tex]\[ P(\text{Def}) = 0.03 + 0.015 + 0.015 \][/tex]
[tex]\[ P(\text{Def}) = 0.06 \][/tex]
Now, we need to find the conditional probability that a defective lens was made by Greens, [tex]\( P(\text{Greens} | \text{Def}) \)[/tex]. To do this, we'll use Bayes' theorem:
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{P(\text{Def} | \text{Greens}) \cdot P(\text{Greens})}{P(\text{Def})} \][/tex]
Substituting in the values:
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{0.05 \cdot 0.60}{0.06} \][/tex]
[tex]\[ P(\text{Greens} | \text{Def}) = \frac{0.03}{0.06} \][/tex]
[tex]\[ P(\text{Greens} | \text{Def}) = 0.50 \][/tex]
Now we need to compare this result with the given choices to determine the correct answer. The choices are:
A. 0.24
B. 0.60
C. 0.71
D. 0.83
The closest value to our calculated probability, 0.50, in the given options is:
B. 0.60
Therefore, the correct answer is:
[tex]\( \boxed{2} \)[/tex]
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