A bookshelf has 5 novels, 3 dictionaries, and 2 encyclopedias. If two books are picked at random, what is the probability that one is a novel and the other is a dictionary?

Official Answer

Certainly! To tackle this probability question, let's first consider the total number of books on the shelf. We have 5 novels, 3 dictionaries, and 2 encyclopedias, giving us a total of 10 books.

When we're looking at the probability of picking one novel and one dictionary randomly, we're essentially dealing with a two-step event. In the first step, we select one book out of the total, and in the second step, we select another book from the remaining ones. The total number of ways to pick any two books from the 10 is a combination problem, calculated as (C(10,2)), which equals (\frac{10!}{2!(10-2)!} = 45). This represents all possible pairs of books we could pick.

Now, let's focus on the specific event of picking one novel and one dictionary. Since there are 5 novels, the number of ways to pick a novel is 5. Similarly, with 3 dictionaries, the number of ways to pick a dictionary is 3. Therefore, the number of ways to pick one novel AND one dictionary is simply the product of these two, which is (5 \times 3 = 15).

To find the probability of this specific event happening, we divide the number of favorable outcomes (picking one novel and one dictionary) by the total number of outcomes (picking any two books). This gives us (\frac{15}{45}), which simplifies to (\frac{1}{3}).

So, the probability that if two books are picked at random, one is a novel and the other is a dictionary, is (\frac{1}{3}).

In my experience, particularly during my time as a Data Scientist, breaking down complex problems into smaller, manageable parts has always been key to finding a solution. This approach not only simplifies the analysis but also allows us to apply fundamental statistical principles effectively. It’s a strategy that has served me well in various projects, including predictive modeling and data analysis, where understanding the underlying probabilities can significantly impact the outcomes.

This methodology offers a flexible framework for approaching probability questions. By dissecting the problem, calculating the total and favorable outcomes, and understanding how they relate, this approach can be adapted and applied to a wide range of scenarios, making it a powerful tool in your interview toolkit.