If a bag contains 8 orange, 7 pink, and 5 violet candies, what is the probability of drawing an orange candy, then a violet candy without replacement?

Official Answer

Certainly! When it comes to tackling probability questions, especially in the realm of data science, it's pivotal to approach them with a clear, logical methodology. Let's break down the given problem step by step, illustrating not just the solution but the thought process behind it, which is critical in data-driven roles.

To start with, we're presented with a scenario where there's a bag containing a mix of candies: 8 orange, 7 pink, and 5 violet, summing up to a total of 20 candies. The task is to calculate the probability of drawing an orange candy followed by a violet candy, without replacement. This implies that once a candy is drawn, it's not put back into the bag, affecting the probability of subsequent draws.

The first step is to determine the probability of drawing an orange candy. Since there are 8 orange candies out of a total of 20, the probability of drawing an orange candy on the first draw is 8/20 or simplified, 2/5.

Next, we move on to the second part of the question, which involves drawing a violet candy immediately after drawing an orange one, without replacement. It's crucial here to recognize that the total number of candies in the bag has now decreased to 19, and importantly, the number of violet candies remains the same since we've only removed an orange candy. Therefore, the probability of drawing a violet candy after an orange one has been drawn and not replaced is 5/19.

To find the combined probability of both these events happening in succession, we multiply the probabilities of each independent event. This is a fundamental principle in probability theory when dealing with sequential events. Thus, the combined probability is (2/5) * (5/19) = 10/95, which simplifies to 2/19.

This calculation illustrates not just a proficiency in handling probability questions, but a systematic approach to problem-solving, breaking down complex scenarios into manageable parts. In the context of data science, this methodology is invaluable, whether it's in analyzing data sets, modeling probabilities, or extracting insights from vast amounts of information.

In summary, the probability of drawing an orange candy followed by a violet candy, without replacement, from the given bag is 2/19. This approach underscores the importance of logical reasoning and methodical problem-solving, hallmarks of effective data analysis and decision-making processes in data science roles.