A box contains 15 electronic components, of which 5 are defective. If 3 components are randomly selected, what is the probability that at least one is defective?

Official Answer

Certainly! When tackling this kind of question, it's pivotal to approach it with a clear method, particularly when you're coming from a Data Scientist background like mine. The richness of experience in handling data, probability, and statistical analysis really comes into play here, providing a robust foundation to address this problem. So, let's dive into it.

Firstly, it's often more straightforward to calculate the probability of the complementary event — in this case, that none of the components selected are defective — and then subtract this value from 1 to find the probability of our event of interest (at least one defective component). This technique is not only efficient but also minimizes the room for error, a strategy I've frequently employed in data analysis to ensure accuracy and insightfulness in my findings.

To calculate the probability that none of the selected components are defective, we start by recognizing that there are 10 good components out of the total 15. The probability that the first component selected is not defective is therefore 10/15. Assuming that happens, we have 9 good components left out of 14 total components for the second draw, making the probability 9/14. Similarly, for the third draw, the probability is 8/13, given the previous two selected were not defective.

Multiplying these probabilities together gives us the overall probability of selecting 3 non-defective components in a row:

(P(\text{none defective}) = \frac{10}{15} \times \frac{9}{14} \times \frac{8}{13})

This calculation simplifies to:

(P(\text{none defective}) = \frac{10 \times 9 \times 8}{15 \times 14 \times 13})

(P(\text{none defective}) = \frac{720}{2730} = \frac{8}{30} = \frac{4}{15})

Now, to find the probability of selecting at least one defective component, we subtract the above result from 1:

(P(\text{at least one defective}) = 1 - P(\text{none defective}))

(P(\text{at least one defective}) = 1 - \frac{4}{15} = \frac{11}{15})

Therefore, the probability that at least one of the three components randomly selected from the box is defective is (\frac{11}{15}).

This approach not only demonstrates a solid grasp of probability concepts but also highlights the analytical mindset I bring to the table — breaking down complex problems into manageable parts, applying mathematical principles, and driving towards accurate, insightful outcomes. This method can be adapted to various scenarios in data science, showcasing the flexibility and depth of analytical skills I'd bring into any role or challenge.