In a bag of 50 balls, 15 are red, 20 are blue, and 15 are green. If 5 balls are drawn at random without replacement, what is the probability that exactly 3 of them are red?

Official Answer

Certainly! In approaching this probability question, I'd like to draw on my background as a Data Scientist, where statistical analysis and probability are integral to interpreting data and predicting outcomes. My experience in crafting algorithms and models to sift through large datasets and derive meaningful insights has honed my skills in probability and combinatorics, which are directly applicable to this question.

To determine the probability of drawing exactly 3 red balls from a bag of 50 balls—where 15 are red, 20 are blue, and 15 are green—when drawing 5 balls without replacement, we delve into combinations. The strategy here involves breaking down the problem into manageable parts and applying the combination formula to calculate the number of ways to achieve the desired outcome.

First, let's calculate the number of ways to draw 3 red balls from the 15 available. This is a straightforward application of the combination formula, which is C(n, k) = n! / [k! * (n-k)!], where n is the total number of items, k is the number of items to choose, and ! denotes factorial.

For the red balls, C(15, 3) = 15! / [3! * (15-3)!] = 455.

Next, since we're drawing 5 balls in total and we want exactly 3 to be red, the remaining 2 must be drawn from the 35 non-red balls (20 blue + 15 green).

For the non-red balls, C(35, 2) = 35! / [2! * (35-2)!] = 595.

To find the total number of ways to draw 5 balls from the 50 without any color restrictions, we use the combination formula again:

For any 5 balls, C(50, 5) = 50! / [5! * (50-5)!] = 2,118,760.

The probability of drawing exactly 3 red balls is the number of favorable outcomes (drawing 3 reds and 2 non-reds) divided by the total number of possible outcomes (any 5 balls from the 50).

Thus, the probability = [C(15, 3) * C(35, 2)] / C(50, 5) = (455 * 595) / 2,118,760 ≈ 0.064.

In this calculation, I've demonstrated not only my ability to navigate complex probability problems but also how such analytical thinking can be applied in data science to dissect and analyze data with precision. Tailoring models to predict specific outcomes or to understand the likelihood of certain events requires a deep understanding of probability and statistics, much like solving this problem. This approach, combined with my experience in data analysis, allows for a methodical and effective way to tackle challenges, predict trends, and make data-driven decisions in a dynamic environment.