Understanding Binomial Probability in a Practical Context

Binomial probability is a fundamental concept in statistics and is often used to model scenarios where there are exactly two possible outcomes, such as whether a bulb is defective or not. In this article, we will explore a practical example where 10 bulbs are manufactured, and 30% of them are defective. We will calculate the probability that all three randomly drawn bulbs are non-defective.

Key Concepts and Definitions

Binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters. It is often applied in situations where we have a fixed number of independent trials, each with only one outcome, and the probability of success is the same for all trials. In the case of bulb production, we have a certain number of bulbs, and each is either defective or not defective.

Example Scenario

Consider a company that manufactures 10 bulbs, with 30% of them being defective. If we randomly draw 3 bulbs without replacement, what is the probability that all three are non-defective?

Solution by Multiple Methods

Let's explore this problem through different methods and compare the results:

Method 1: Sequential Probability Calculations

Here, we calculate the probability of drawing three non-defective bulbs one at a time:

First Attempt: The probability of picking a non-defective bulb first is (10 - 3) / 10 7/10 0.7. Second Attempt: Given one non-defective bulb is already picked, the probability of the second bulb also being non-defective is (9 - 3) / 9 6/9 0.6667. Third Attempt: Similarly, the probability of the third bulb being non-defective is (8 - 3) / 8 5/8 0.625.

The combined probability of all three events happening is:

$$ frac{7}{10} times frac{6}{9} times frac{5}{8} frac{7 times 6 times 5}{10 times 9 times 8} frac{210}{720} frac{7}{24} approx 0.2917 $$

Method 2: Combinatorial Approach

We can use the binomial coefficient to find the number of ways to choose 3 non-defective bulbs out of 7 non-defective bulbs:

$$ binom{10}{3} times 0.7^3 times 0.3^7 $$

The binomial coefficient (binom{10}{3}) is the number of ways to choose 3 non-defective bulbs out of 10, which is 120. Therefore, the probability is:

$$ frac{120 times 0.3^3 times 0.7^7}{1 times 1} 0.2668 $$

Method 3: Simplified Approach

Another straightforward method is to calculate it directly:

$$ frac{7}{10} times frac{6}{9} times frac{5}{8} 0.2917 $$

Evaluation of Different Solutions

Let's compare the different solutions:

The first method provides a probability of 0.2917 or 29.17%. The combinatorial approach gives a probability of 0.2668 or 26.68%.

The slight discrepancy between the two methods can be attributed to the nuances of combinatorial calculations and probability rules. Both approaches are valid, but the second method using binomial coefficients is more aligned with theoretical probability.

Conclusion and Insights

In conclusion, the probability of drawing three non-defective bulbs from 10 bulbs, where 30% are defective, is approximately 26.68%. This example demonstrates the application of binomial distribution in practical scenarios and highlights the importance of understanding various problem-solving methods in statistics.

Related Keywords

Binomial Distribution Probability Calculations Non-Defective Bulbs

References

For further reading and understanding, refer to:

MIT OpenCourseWare - Probability and Statistics Statistics How To - Binomial Distribution