Probability of Drawing Defective or Non-Defective Bulbs

Probability is a fundamental concept in mathematics and statistics, helping us understand the likelihood of certain outcomes. In this article, we will delve into the probability of drawing defective and non-defective bulbs from a stock, and how these probabilities can be calculated using basic and advanced approaches.

Basic Probability Concepts

When selecting bulbs from a stock, several scenarios can arise depending on the condition of the bulbs. Let's explore the basic probability calculations related to these scenarios.

Scenario 1: 8 Electric Bulbs in Stock with 3 Defective Ones

Consider a shop that has a stock of 8 electric bulbs out of which 3 are defective. A customer demands two bulbs. A shopkeeper picks up two bulbs randomly. We aim to find the probability that one bulb is defective and the other is not defective.

Calculation

Firstly, let's understand the total number of ways to pick 2 bulbs from 8:

[ C_8^2 frac{8!}{2!(8-2)!} frac{8 times 7}{2 times 1} 28 ]

Next, we calculate the number of ways to pick 1 defective and 1 non-defective bulb. There are 3 defective bulbs, and we need to choose 1 out of these 3:

[ C_3^1 3 ]

There are 5 non-defective bulbs, and we need to choose 1 out of these 5:

[ C_5^1 5 ]

The number of favorable outcomes is the product of these two combinations:

[ 3 times 5 15 ]

Therefore, the probability of picking one defective and one non-defective bulb is:

[ frac{15}{28} ]

Scenario 2: 100 Bulbs with 8 Defective Ones

Another scenario involves a set of 100 bulbs, out of which 8 are defective. We need to find the probability of drawing a defective or non-defective bulb.

Calculation

Firstly, the number of non-defective bulbs is:

[ 100 - 8 92 ]

The probability of drawing a non-defective bulb is:

[ P(text{non-defective}) frac{92}{100} frac{23}{25} ]

The probability of drawing a defective bulb is:

[ P(text{defective}) frac{8}{100} frac{2}{25} ]

Advanced Probability Distribution

When dealing with more advanced scenarios, such as the number of defective bulbs in a subset, we can use the hypergeometric distribution.

Hypergeometric Distribution

The hypergeometric distribution is used for sampling without replacement. The formula for the hypergeometric distribution is:

[ P(X k) frac{binom{K}{k} binom{N-K}{n-k}}{binom{N}{n}} ]

where ( N ) is the population size, ( K ) is the number of success states in the population, ( n ) is the number of draws, and ( k ) is the number of successes in the draws.

Example Calculation

Using the hypergeometric distribution to find the probability of drawing exactly one defective bulb from a batch of 10 bulbs (2 defective and 8 non-defective) is as follows:

[ P(text{exactly one defective}) frac{binom{2}{1} binom{8}{1}}{binom{10}{2}} ]

Firstly, calculate the combinations:

[ binom{2}{1} 2 ] [ binom{8}{1} 8 ] [ binom{10}{2} frac{10 times 9}{2 times 1} 45 ]

The probability is then:

[ P(text{exactly one defective}) frac{2 times 8}{45} frac{16}{45} approx 0.356 ]

Similarly, for exactly 2 defective bulbs:

[ P(text{exactly two defective}) frac{binom{2}{2} binom{8}{0}}{binom{10}{2}} ]

Here, (binom{2}{2} 1) and (binom{8}{0} 1), so:

[ P(text{exactly two defective}) frac{1 times 1}{45} frac{1}{45} approx 0.022 ]

Conclusion

Understanding probability in the context of defective and non-defective bulbs is crucial for various applications, such as quality control and inventory management. By using basic and advanced probability calculations, we can make informed decisions in such scenarios.

References

[Reference 1]: Basic concepts of probability. Link to Reference

[Reference 2]: Hypergeometric distribution. Link to Reference