1. សិប្បកម្មមួយមានម៉ាស៊ីន 2 ផលិតវត្ថុដូចគ្នា។ គេដឹងថាម៉ាស៊ីន ទី1 ផលិតវត្ថុបាន 45% ដែលក្នុងនោះ មានខូច 3% និង ម៉ាសុីនទី 2 ផលិតបាន វត្ថុ បាន 55% ដែលក្នុងនោះ មានខូច 2% ។ 
 ក. តើប្រូបាប ដែលមានវត្ថុខូច ដោយដឹងថា ផលិតផលម៉ាស៊ីន ទី 1 ស្មើនឹងប៉ុន្មាន? 
 ខ. រកប្រូបាបដែលមានវត្ថុខូចក្នុងម៉ាសុីនទី 1
គ. រកប្រូបាបដែលខូចក្នុងម៉ាសុីនទី2 ។ 
 ឃ. បើម៉ាសុីនទី2 ផលិតបានចំនួន 1000  តើម៉ាសុីនទី1 ខូចអស់ប៉ុន្មាន? ម៉ាសុីនទី 2 ខូចប៉ុន្មាន ? 
 Please tell me 

To solve the given problems, we need to analyze the probabilities and quantities related to the production and defect rates of the two machines. Here's the step-by-step breakdown:

**Given:**
1. Machine 1 produces 45% of the total products, with a defect rate of 3%.
2. Machine 2 produces 55% of the total products, with a defect rate of 2%.

Let's denote:
- \( P(M1) \) as the probability that a product is produced by Machine 1.
- \( P(D|M1) \) as the probability that a product is defective given that it was produced by Machine 1.
- \( P(M2) \) as the probability that a product is produced by Machine 2.
- \( P(D|M2) \) as the probability that a product is defective given that it was produced by Machine 2.

From the problem:
- \( P(M1) = 0.45 \)
- \( P(D|M1) = 0.03 \)
- \( P(M2) = 0.55 \)
- \( P(D|M2) = 0.02 \)

Now let's solve each part:

### ក. Probability that a product is defective given it was produced by Machine 1
This is given directly as \( P(D|M1) = 0.03 \) or 3%.

### ខ. Probability of a product being defective in Machine 1
The overall probability of a product being defective from Machine 1 can be calculated by multiplying the probability of a product being made by Machine 1 and the probability of a product being defective if it was made by Machine 1.

\[ P(D \cap M1) = P(M1) \times P(D|M1) \]
\[ P(D \cap M1) = 0.45 \times 0.03 = 0.0135 \]

So, the probability that a product is defective and produced by Machine 1 is 0.0135 or 1.35%.

### គ. Probability of a product being defective in Machine 2
Similarly, the probability of a product being defective from Machine 2 is:

\[ P(D \cap M2) = P(M2) \times P(D|M2) \]
\[ P(D \cap M2) = 0.55 \times 0.02 = 0.011 \]

So, the probability that a product is defective and produced by Machine 2 is 0.011 or 1.1%.

### ឃ. Number of defective products if Machine 2 produces 1000 units

If Machine 2 produces 1000 units, we can find the defective products for both machines proportionally.

First, let's find out the total number of products produced by Machine 1 if Machine 2 produces 1000 units. Since Machine 2 produces 55% of the products, we can set up a proportion to find the total number of products:

\[ 0.55 \times \text{Total Products} = 1000 \]
\[ \text{Total Products} = \frac{1000}{0.55} \approx 1818.18 \]

So, the total number of products is approximately 1818.

Now, the number of products produced by Machine 1 is:

\[ \text{Number of products from Machine 1} = 0.45 \times 1818 \approx 818.18 \]

Now, calculate the number of defective products from each machine:
- Machine 1: 
\[ 818.18 \times 0.03 \approx 24.55 \]

- Machine 2: 
\[ 1000 \times 0.02 = 20 \]

So, approximately 25 products from Machine 1 and 20 products from Machine 2 are defective.

**Summary:**
1. \( P(D|M1) = 0.03 \) or 3%.
2. \( P(D \cap M1) = 0.0135 \) or 1.35%.
3. \( P(D \cap M2) = 0.011 \) or 1.1%.
4. If Machine 2 produces 1000 units, Machine 1 produces about 818 units, with approximately 25 defective products from Machine 1 and 20 defective products from Machine 2.

Would you like more details or have any questions?

### Additional Questions:
1. What is the overall probability of a product being defective?
2. How would the calculations change if Machine 1's defect rate changed to 4%?
3. How many total products would be defective if both machines produced 2000 units each?
4. What is the probability of a product being produced by Machine 2 and being non-defective?
5. If the total number of defective products is known, can we determine the number produced by each machine?
6. How would you model this problem using a tree diagram?
7. What is the probability of selecting a defective product from a mixed batch of 1000 units?
8. How do the production probabilities affect the overall defect rate?

### Tip:
When working with probabilities, clearly defining and understanding the given conditions and how they relate to each other is crucial for accurate calculations.

Explore the analysis of machine production and defect rates, including probabilities and quantities related to two machines producing similar products with different defect rates. Get detailed step-by-step solutions and calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation