# Probability of defective machines question

• zhfs
In summary, the conversation discusses a probability question involving three machines in a factory that produce a certain percentage of items and have a certain percentage of defective items. The question asks for the probability of selecting a defective item and the probability that a defective item was produced by machine A. The conversation ends with someone providing a solution using Bayes' theorem and another person providing a different approach using an example with 1000 items. Both methods result in the same answer for part (a) and part (b).

#### zhfs

[solved]probability question

## Homework Statement

Three machines A, B, and C produce 50%, 30% and 20% respectively of the total
number of items at a factory. Each produces a number of defective: 3%, 4% and 5%
respectively.
(a) If an item produced by one of these machines is selected, find the probability that
it is defective.
(b) Now suppose that a manufactured item is selected at random and is found to be
defective. Find the probability that this item was produced by machine A.

N/A

## The Attempt at a Solution

is part (a)
Pr = (3%+4%+5%)/(50%*30%*20%)?

and no idea on part b

thanks a lot!

Last edited:
Use Bayes' theorem.

Let D be the event that a selected item is defective

Let A, B, and C be the events that that an item is produced by machine A, B, and C respectively

$$P(D) = P(D\cap A) + P(D \cap B) +P(D \cap C)$$

$$= P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)$$

$$P(A|D) = \frac {P(A \cap D)}{P(D)}$$

$$= \frac {P(D|A)P(A)}{P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)}$$

How I would do it: Imagine that 1000 of the items are produced. Then 50% of 1000= 500 come from machine A, 30% of 1000= 300 from machine B, and 20% of 1000= 200 from machine C.

Of the 500 from machine A, 3% of 500= 15 are defective, 4% of 300= 12 are from machine B, and 2% of 200= 4 are from machine C.

A total of 15+ 12+ 4= 41 defective items are produced of a total of 1000 items. With that information (a) is easy.

Of a total of 41 defective items, 15 of them were from A. With that information, (b) is easy.

thank you guys!

## 1. What is the probability of a machine being defective?

The probability of a machine being defective can be calculated by dividing the number of defective machines by the total number of machines in a given population or sample. This is known as the probability of a single event. For example, if there are 10 defective machines out of 100 total machines, the probability of a machine being defective is 10/100 or 0.1 (10%).

## 2. How do you determine the probability of multiple defective machines in a population?

The probability of multiple defective machines in a population can be calculated using the binomial probability formula. This formula takes into account the number of trials (machines), the probability of success (defectiveness), and the desired number of successes (defective machines). For example, if we wanted to find the probability of having 2 defective machines out of 100, we would use the formula P(x=2) = (100 choose 2) * (0.1)^2 * (0.9)^98 = 0.1937 or 19.37%.

## 3. Can the probability of a defective machine change over time?

Yes, the probability of a machine being defective can change over time. This can be due to a variety of factors such as wear and tear, maintenance, and human error. It is important to regularly monitor the probability of defective machines in order to identify any changes and address any issues that may arise.

## 4. How does the sample size affect the probability of defective machines?

The sample size can have an impact on the probability of defective machines. As the sample size increases, the probability of having a defective machine in the sample also increases. This is because a larger sample size allows for a more accurate representation of the overall population. Therefore, it is important to have a sufficiently large sample size in order to accurately estimate the probability of defective machines in a population.

## 5. How can probability be used to improve the quality of machines?

Probability can be used to improve the quality of machines in several ways. Firstly, it can be used to identify potential issues or areas of concern in the production process, allowing for targeted improvements to be made. Additionally, probability can be used to set quality control standards and determine acceptable levels of defectiveness. By regularly monitoring and analyzing the probability of defective machines, improvements can be made to increase the overall quality and reliability of machines.

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