A box contains 10 bulbs of which 3 are defective. If 2 bulbs are chosen at random, what is the probability that both are defective?

Official Answer

Given the scenario where we have a box of 10 bulbs, out of which 3 are defective, and we're tasked with selecting 2 bulbs at random, the question at hand is to determine the probability that both bulbs we pick are defective. Now, approaching this problem, we'll employ fundamental principles of probability, specifically focusing on combinations, to calculate our desired outcome. This method is not only efficient but also provides a clear, logical structure for tackling similar problems, which is particularly useful in my role as a Data Scientist.

To start, let's consider the total number of ways we can pick 2 bulbs from the 10 available. This is essentially a combination problem, where order does not matter, and we're selecting 2 out of 10. The formula for combinations is given by (C(n, k) = \frac{n!}{k!(n-k)!}), where (n) is the total number of items, (k) is the number of items to choose, and (n!) denotes the factorial of (n). Applying this formula, we find the total number of ways to choose 2 bulbs from 10 is (C(10, 2) = \frac{10!}{2!(10-2)!} = 45).

Next, we focus on the specific case of picking 2 defective bulbs out of the 3 available. Applying the combination formula again, we find the number of ways to choose 2 defective bulbs from 3 is (C(3, 2) = \frac{3!}{2!(3-2)!} = 3).

Now, the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. In our scenario, the favorable outcomes are the number of ways to pick 2 defective bulbs (which we found to be 3), and the total number of possible outcomes is the total number of ways to pick any 2 bulbs from the 10 (which is 45). Therefore, the probability that both bulbs picked are defective is (\frac{3}{45}), which simplifies to (\frac{1}{15}).

This approach, grounded in the principles of probability and combinations, not only helps solve the problem at hand but also equips us with a robust framework for analyzing similar problems. In my role, whether analyzing data trends, optimizing algorithms, or predicting outcomes, the ability to dissect and solve problems through a fundamental mathematical lens is invaluable. Furthermore, this methodology is adaptable, allowing for nuanced adjustments based on the specifics of the problem, making it a powerful tool in the arsenal of a Data Scientist.