Question 1169525: A previous survey shows that a machine making plastic components is correctly set
up for the day's production on 85% of days. On days when it is set up correctly, 96%
of the components produced are good. If the machine is not set up correctly, only 30%
of the components produced are good. On a particular day, the machine is set up and
the first component produced is found to be good. What is the probability that the
machine is set up correctly?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem using Bayes' Theorem.
**Define the Events**
* **C:** The machine is set up correctly.
* **G:** The first component produced is good.
**Given Probabilities**
* $P(C) = 0.85$ (Probability the machine is set up correctly)
* $P(C') = 1 - P(C) = 1 - 0.85 = 0.15$ (Probability the machine is not set up correctly)
* $P(G|C) = 0.96$ (Probability of a good component given the machine is set up correctly)
* $P(G|C') = 0.30$ (Probability of a good component given the machine is not set up correctly)
**We need to find:**
* $P(C|G)$ (Probability the machine is set up correctly given the component is good)
**Apply Bayes' Theorem**
Bayes' Theorem states:
$$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$
Where $P(G)$ can be found using the law of total probability:
$$P(G) = P(G|C) \cdot P(C) + P(G|C') \cdot P(C')$$
**Calculations**
1. **Calculate $P(G)$:**
$$P(G) = (0.96 \times 0.85) + (0.30 \times 0.15)$$
$$P(G) = 0.816 + 0.045$$
$$P(G) = 0.861$$
2. **Calculate $P(C|G)$:**
$$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$
$$P(C|G) = \frac{0.96 \times 0.85}{0.861}$$
$$P(C|G) = \frac{0.816}{0.861}$$
$$P(C|G) \approx 0.9477$$
**Final Answer**
The probability that the machine is set up correctly given that the first component produced is good is approximately 0.9477.
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
A previous survey shows that a machine making plastic components is correctly set
up for the day's production on 85% of days. On days when it is set up correctly, 96%
of the components produced are good. If the machine is not set up correctly, only 30%
of the components produced are good. On a particular day, the machine is set up and
the first component produced is found to be good. What is the probability that the
machine is set up correctly?
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The problem's formulation in the post is INCORRECT.
In this formulation, the problem can not be solved/answered.
To be correct, it should be re-formulated, re-edited and re-written
from scratch.
By reading this problem formulation, I clearly see that a person
who created it is mathematically incompetent in the subject.
The "solution" in the post by the other tutor @CPhill is GIBBERISH
produced by an undertrained artificial intelligence.
To keep you mind in safe condition, I recommend you to ignore
both the problem and its quasi-"solution" produced by @CPhill.
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Regarding the post by @CPhill . . .
Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.
The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.
Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.
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