When conducting a random ball selection from a sack containing balls of various colors, understanding the probability of extracting specific combinations is crucial. In this article, we will delve into the intricate steps of calculating the probability of selecting 2 white balls, 1 black ball, and 1 grey ball without replacement. This example serves as a fundamental yet powerful tool in probability theory and its application in real-world scenarios. Let's explore the detailed process.

Problem Definition

We start with a sack containing 10 balls, with the following distribution:

The task is to calculate the probability of selecting 4 balls in the following sequence: 2 white, 1 black, and 1 grey, without replacement.

Step-by-Step Solution

Let's break down the process of selecting the balls step by step, adhering to the given sequence of 2 white, 1 black, and 1 grey.

Step 1: Drawing the First White Ball

The probability of drawing the first white ball from 10 total balls, where 3 are white, is:

[frac{3}{10}]

Step 2: Drawing the Second White Ball

With the first white ball drawn and without replacement, there are now 9 balls left in the sack, with 2 white balls remaining. The probability of drawing the second white ball is:

[frac{2}{9}]

Step 3: Drawing the Black Ball

After drawing the second white ball, 8 balls remain in the sack, with 3 black balls still available. The probability of drawing a black ball is:

[frac{3}{8}]

Step 4: Drawing the Grey Ball

With 7 balls left in the sack, and 2 grey balls remaining, the probability of drawing a grey ball is:

[frac{2}{7}]

Calculating the Combined Probability

The overall probability of successfully drawing the balls in the specified sequence is the product of the individual probabilities:

[left(frac{3}{10}right) times left(frac{2}{9}right) times left(frac{3}{8}right) times left(frac{2}{7}right) frac{36}{5040} frac{1}{140} approx 0.0071]

This detailed step-by-step process and calculation reveal the probability of the specified sequence of ball draws.

Conclusion

By understanding and performing such calculations, we can gain a deeper insight into probability theory and its practical applications in various fields. Whether it's in statistical analysis, gambling, or even in certain algorithms and simulations, the ability to calculate probabilities accurately is invaluable. This example demonstrates how to handle such scenarios with precision and provides a clear methodology for future reference.