Ensuring at Least Two Balls of the Same Color: A Pigeonhole Principle Application

When dealing with a bag containing balls of two different colors, the question arises: what is the minimum number of balls one must pick to ensure that at least two balls are of the same color? This problem can be elegantly solved using the Pigeonhole Principle. Let's delve into the details.

Understanding the Pigeonhole Principle

The Pigeonhole Principle is a fundamental concept in combinatorics, which states that if you have more items than containers, at least one container must contain more than one item. In our case, the containers are the colors of the balls.

Application to the Problem

Let's consider a bag containing balls of two different colors, denoted as A and B. The objective is to determine the minimum number of balls that need to be picked to ensure at least two balls of the same color.

Calculation Using the Pigeonhole Principle

If we pick 1 ball, it could be either color, say A or B. If we pick 2 balls, they could still be of different colors - one A and one B. However, if we pick 3 balls, by the Pigeonhole Principle, at least one of the colors must appear at least twice. Therefore, the minimum number of balls to be picked is 3.

Interesting Exception

Although the likelihood is exceptionally low, it is theoretically possible to pick all the balls of one color before selecting a ball of the other color. To ensure that at least one ball of each color is selected, the number of balls drawn must be one more than the smaller number of balls of one color. Thus, if there are at least 3 balls of each color, you need to pick 4 balls to guarantee at least one ball of each color.

Example with 3 Balls of Each Color

Consider a scenario where there are 3 balls of color A and 3 balls of color B. If only 2 balls are selected, the possible combinations are AA, BB, and AB (BA is the same as AB). There are only 4 possible permutations, and only 2 of them have at least two balls of the same color. Therefore, the criterion of having at least 2 balls of the same color is not met.

However, if 3 balls are selected, the possible combinations include AAA, BBB, AAB, ABA, BAA, BBA, BAB, and ABB. There are 8 permutations, and all of them have at least two balls of the same color. Consequently, the minimum number of balls required to ensure at least two balls of the same color is 3.

Conclusion

By using the Pigeonhole Principle and combinatorial mathematics, we can derive the minimum number of balls needed to ensure at least two balls of the same color. In the case where there are at least 3 balls of each color, the answer is unequivocally 3. This application of the Pigeonhole Principle is crucial in various fields, from computer science to probability theory, where understanding and predicting such outcomes is essential.