Combinations and Probability: Selecting Balls from a Box with Conditions

This article examines the process of selecting ping pong balls from a box under specific conditions. Specifically, we will calculate the number of ways to select 5 balls such that one is green and four are orange. We will also explore related concepts such as combinations, probability, and distinguishability of balls.

Introduction

Imagine a box containing a variety of ping pong balls: 5 green, 7 yellow, and 10 orange. We are tasked with finding the number of ways to select 5 balls such that one is green and four are orange. This problem can be approached using the principles of combinations and probability, which are fundamental concepts in statistics and combinatorics.

Understanding the Problem

The task is to select 5 balls, with the condition that one is green and four are orange. To solve this, we can break down the problem into two parts: selecting one green ball and four orange balls separately, and then combining the results.

Step 1: Selecting One Green Ball

From the 5 green balls available, we need to select one. This can be done using the combination formula:

C(n, r) n! / (r!(n-r)!)

Plugging in the values for our scenario:

C(5, 1) 5! / (1!(5-1)!) 5! / (1! * 4!) 5

Step 2: Selecting Four Orange Balls

Now, we need to select 4 orange balls out of the 10 available. Using the combination formula again:

C(10, 4) 10! / (4!(10-4)!) 10! / (4! * 6!) (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) 210

Total Number of Ways

To find the total number of ways to select 5 balls such that one is green and four are orange, we multiply the number of ways to select the green ball by the number of ways to select the orange balls:

Total ways C(5, 1) * C(10, 4) 5 * 210 1050

Therefore, there are 1050 ways to select 5 balls from the box such that one is green and four are orange.

Discussion and Application

This problem involves the use of combinations, which is a key concept in probability and statistics. Combinations are used to calculate the number of ways to choose a subset from a larger set when the order of selection does not matter.

Indistinguishable vs. Distinguishable Balls

The problem can be further analyzed based on whether the balls are distinguishable or not. For example:

Indistinguishable yellow balls: There is only 1 way to choose 7 yellow balls since they are all the same. Distinguishable but order doesn't matter: We would use the combination formula as done earlier. Distinguishable and order matters: We would use permutations, which is a different calculation.

Probability Calculation

If we want to calculate the probability of this event occurring, we can use the formula:

P(A) Number of favorable outcomes / Total number of possible outcomes

From the previous calculations, we know:

Number of favorable outcomes 1050 Total number of possible outcomes: C(22, 5) 22! / (5!(22-5)!) 26334

Therefore:

P(A) 1050 / 26334 0.0399

Conclusion

The process of solving such a problem involves understanding combinations and probability. The result of 1050 ways to select 5 balls, one green and four orange, is a clear application of these concepts. The discussion on the distinguishability of the balls shows the importance of considering the nature of the objects when solving such problems.

References

Park, J. (2021). Introduction to Combinatorics and Probability. United States: Wiley. Moore, D. S., McCabe, G. P. (2003). Introduction to the Practice of Statistics. New York: W.H. Freeman and Company.

Further Reading

For a deeper dive into the concepts of combinations and probability, you might want to explore the following resources:

Combinations Formula in Probability Combinations vs Permutations Combination and Permutations Explained