A certain brand of light bulb has a failure rate of 2% when tested. If 100 bulbs are tested, what is the probability that exactly 5 bulbs will fail?

Official Answer

To tackle this question, we'll use the binomial probability formula since we're dealing with a discrete number of trials (testing 100 bulbs) and we're interested in finding the probability of a specific number of successes (in this case, failures, oddly enough, which are our 'success' criteria here). The failure rate, or our probability of success on any given trial, is 2% or 0.02. So, we're essentially looking for the probability of exactly 5 failures out of 100 trials.

The binomial probability formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successful trials we're interested in, 'p' is the probability of success on any given trial, and '(n choose k)' is the binomial coefficient, calculated as n! / (k!(n-k)!).

Substituting our values into this formula, we get P(X = 5) = (100 choose 5) * 0.02^5 * (1-0.02)^(100-5). Calculating '(100 choose 5)' gives us 75,287,520 ways to choose 5 failures out of 100 trials. Multiplying this by 0.02^5 gives us the probability of having these 5 failures in any specific arrangement of 100 trials. Multiplying further by (1-0.02)^(100-5) accounts for the probability of the remaining 95 bulbs not failing.

Computing this, we find that P(X = 5) is approximately equal to 0.0815, or 8.15%. This means, if we were to test 100 bulbs, there's about an 8.15% chance exactly 5 of them will fail.

In a professional context, especially in roles such as Data Scientist or Data Analyst, it's crucial to understand and apply statistical principles to analyze data and predict outcomes. This question, while straightforward, tests the ability to apply statistical formulas in practical scenarios. The ability to compute and interpret such probabilities can significantly impact decision-making processes, from predicting failure rates in production environments to assessing risks in new projects.

Moreover, this framework isn't just about solving a probability question; it's a testament to a systematic approach to problem-solving, emphasizing the importance of understanding underlying principles. Whether you're preparing data for analysis, building predictive models, or evaluating the outcomes of a business strategy, the principles remain the same. Tailoring this approach in your responses can help illuminate your analytical prowess and your capacity to apply theoretical knowledge in practical, impactful ways.