# Conditional probability and defect rate

• Kinetica
In summary, Line I has a defect rate of 8%, whereas line II has a defect rate of 10%. If an item is chosen at random from the day’s production, find the probability that it will not be defective.
Kinetica

## Homework Statement

Of the items produced daily by a factory, 40% come from line I and 60% from line II. Line I
has a defect rate of 8%, whereas line II has a defect rate of 10%. If an item is chosen at random
from the day’s production, find the probability that it will not be defective.

## The Attempt at a Solution

My answer is 0.82 because Line I with no defect is 0.32+Line II with no defect is 0.50.
OR
1-Both lines defected (0.18)=0.82.

Is my approach correct?

I'd like to know how your numbers came about. Try drawing a tree diagram and see if you can find the results then; it'll help simplify the process.

I drew a table:
Def No Def
I 0.08 0.32 0.40
II 0.10 0.50 0.60
0.18 0.82 1

It is a little bit off but you can see that elements add up to 1. and you can also see the number 0.82 - this is my answer.

No no, I'm requesting that you write out how you got your numbers. I'll give a hint; question if your table is correct. If you have a .4 chance of it being in line 1, and a .6 chance of the item being in line 2, would you say that the chance of it being in line 1 AND defect is .08? Or would it be something else?

I see what you mean. My mistake is that I did not consider to multiply the probabilities by corresponding values.

Then, my question is. Some problems that I solved earlier were easily solved by using tables like this. How do I distinguish which method to use?

That's the tricky part. A key component is how the question is asked, but nothing beats practice. Probability is one of those subjects where you just have to keep doing problems over and over to gain intuition. Best of luck!

-J

Kinetica said:
I see what you mean. My mistake is that I did not consider to multiply the probabilities by corresponding values.

Then, my question is. Some problems that I solved earlier were easily solved by using tables like this. How do I distinguish which method to use?

Table, trees, formulas, whatever---they all say the same thing, perhaps in disguised form. There is no single right way: do whatever makes you fee comfortable.

Let's compute P(D) = prob item is defective.
(1) Formula: P(D) = P(D|I)*P(I) + P(D|II)*P(II) = (.08)*(.60)+(.10)*(.40) = 0.0.088
(2) Table: Say we make 1,000,000 items.
No. produced in Line I = (0.60)(1,000,000) = 600,000
No. produced in Line II = (0.40)(1,000,000) = 400,000
Of the 600,000 produced on LI, the number defective = (.08)(600,000) = 48,000
Of the 400,000 produced on LII the number defective = (.10)(400,000) = 40,000
Putting these in a table we have:

Total Defect Non-defect
Line I items 600,000 48,000 552,000
Line II items 400,000 40 000 360,000
Total 1,000,000 88,000 912,000
Thus, P(D) = 88,000/1,000,000 = 88/1000 = 0.088.

You can also do it in a tree, but I can't easily draw a tree here.

RGV

## 1. What is conditional probability?

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B) and is calculated by dividing the probability of both A and B occurring by the probability of B occurring.

## 2. How is conditional probability used in defect rate analysis?

In defect rate analysis, conditional probability is used to calculate the probability of a defect occurring given that a certain factor or condition is present. This helps in identifying the root cause of defects and making informed decisions to improve defect rates.

## 3. What is the relationship between conditional probability and defect rate?

The relationship between conditional probability and defect rate is that conditional probability is used to calculate the likelihood of a defect occurring, which in turn affects the overall defect rate. By understanding the conditional probabilities of different factors, we can make changes to improve the defect rate.

## 4. Can conditional probability be used to predict future defect rates?

Yes, conditional probability can be used to predict future defect rates by analyzing past data and identifying patterns and relationships between different factors and defects. By using this information, we can make informed predictions about future defect rates and take proactive measures to improve them.

## 5. What are some common challenges in using conditional probability for defect rate analysis?

Some common challenges in using conditional probability for defect rate analysis include having incomplete or inaccurate data, identifying the right factors and conditions to calculate probability for, and understanding the limitations of the data and assumptions made in the analysis process.

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