Probability of Drawing All White Balls from a Bag: A Comprehensive Guide

In this article, we will explore the concept of probability and combinatorics, specifically focusing on the scenario of drawing all white balls from a bag without replacement. We will break down the calculations, provide step-by-step solutions, and discuss various methods to solve similar problems.

Scenario Overview

Imagine a bag that initially contains 5 white, 7 red, and 8 black balls. The question is: What is the probability of drawing 4 white balls one by one without replacement?

Step-by-Step Solution

The probability of drawing 4 white balls from the bag can be calculated by considering the number of ways to draw 4 white balls out of the 5 white balls, divided by the total number of ways to draw 4 balls from the 20 total balls.

Calculation Methods

Direct Calculation

The number of ways to draw 4 white balls out of 5 is given by the combination formula (nCk frac{n!}{k!(n-k)!})

Number of ways to choose 4 white balls out of 5:

[5C4 frac{5!}{4!1!} 5]

The total number of ways to draw 4 balls from the 20 balls is:

[20C4 frac{20!}{4!16!} 4845]

Therefore, the probability of drawing 4 white balls is:

[P frac{5}{4845} frac{1}{969}]

Sequental Drawing Calculation

Another way to look at this problem is by considering the probability of drawing each white ball sequentially without replacement:

Hence, the probability of drawing 4 white balls one after another is:

[frac{5}{20} times frac{4}{19} times frac{3}{18} times frac{2}{17} frac{1}{969}]

Thus, the probability of drawing all 4 balls being white is the same using both methods.

Advanced Scenario: Four Balls of Different Colors

Now consider the situation where there are only 3 colors: white, red, and black. Can 4 balls be drawn from the bag such that each ball is of a different color? Since there are only three colors, it is impossible to have 4 balls of different colors. Therefore, the probability of drawing all 4 balls being of different colors is 0.

General Calculation for Drawing Any Number of White Balls

Assuming 4 balls are drawn sequentially but without replacement, we can calculate the probability of drawing 0 to 4 white balls:

0 white balls: [5C0 times 15C4 / 20C4 1 times 1365 / 4845 frac{1}{3.5}] 1 white ball: [5C1 times 15C3 / 20C4 5 times 455 / 4845 frac{40}{91}] 2 white balls: [5C2 times 15C2 / 20C4 10 times 105 / 4845 frac{30}{91}] 3 white balls: [5C3 times 15C1 / 20C4 10 times 15 / 4845 frac{20}{273}] 4 white balls: [5C4 times 15C0 / 20C4 5 times 1 / 4845 frac{1}{273}]

The probability of "majority white balls" (3 or 4 white) is:

[105 / 1365 frac{1}{13}]

Conclusion

In summary, we have explored the probability of drawing all white balls from a bag with different combinations and methods. This problem serves as a fundamental example in combinatorics and probability, applicable to a wide range of real-world scenarios.