Probability in Quality Control: Defective Cellphones Among Non-Defective Ones

Quality control is a critical component in maintaining the standards of manufacturing processes. This article delves into a specific scenario: a box containing 5 defective and 195 non-defective cellphones. A quality control engineer selectively picks 2 cellphones at random without replacement. This article explains the step-by-step process to find the probability that exactly one of the selected cellphones is defective.

Step-by-Step Probability Calculation

To find the probability that exactly one of the two selected cellphones is defective, we will use the concept of combinations and basic probability. Here’s a detailed breakdown:

Determining the Total Number of Cellphones

Total cellphones Defective cellphones Non-defective cellphones

Given:

Total defective cellphones 5 Total non-defective cellphones 195 Total cellphones 5 195 200

Calculating the Total Ways to Choose 2 Cellphones from 200

The number of ways to choose 2 cellphones from 200 is given by the combination formula:

Calculating the Number of Favorable Outcomes for Exactly One Defective Cellphone

To get exactly one defective cellphone, we need to choose:

1 defective cellphone from 5 defective ones 1 non-defective cellphone from 195 non-defective ones

Number of ways to choose 1 defective cellphone:

Number of ways to choose 1 non-defective cellphone:

Calculating the Total Number of Favorable Outcomes

Using the above results, the total number of ways to choose 1 defective and 1 non-defective cellphone is:

Favorable outcomes (Number of ways to choose 1 defective) × (Number of ways to choose 1 non-defective)

Favorable outcomes 5 × 195 975

Calculating the Probability

The probability of selecting exactly one defective cellphone is given by the ratio of the number of favorable outcomes to the total outcomes:

Simplifying the Fraction

To simplify :

Find the greatest common divisor (GCD) of 975 and 19900. The GCD is 25.

Divide both the numerator and denominator by the GCD:

Final Answer:

The probability that exactly one of the selected cellphones is defective is:

Related Scenarios

A similar probability calculation can be applied to scenarios like:

Probability of selecting two defective items from a lot consisting of 20 defective and 80 non-defective items without replacement. General concepts in quality control and the application of probability in manufacturing and inspection processes.

Conclusion

Understanding the probability of selecting a specific number of defective items in a quality control process is crucial for manufacturers and quality assurance teams. The steps outlined above can be applied to various scenarios involving combinations and basic probability techniques to ensure product quality and reliability.