Combinatorial Analysis of Jelly Bean Samples: Exploring Sample Spaces
This article delves into the combinatorial analysis of a jar containing jelly beans in various colors. By understanding the concept of sample spaces, we can calculate the number of ways to randomly select two jelly beans from the jar. This is crucial for various applications, including probability calculations and statistical analysis.
Introduction
The jar in question contains jelly beans in six different colors: pink, purple, white, black, orange, and green. The quantities of each color are as follows:
Pink: 87 Purple: 74 White: 35 Black: 70 Orange: 25 Green: 47Calculating the Total Number of Jelly Beans
The total number of jelly beans in the jar can be calculated by summing the quantities of each color:
Total 87 74 35 70 25 47 338
Sample Space for Two Jelly Beans
When selecting two jelly beans from the total, we need to determine the size of the sample space. A sample space is the set of all possible outcomes when random selection occurs. Here, we will calculate this using the combination formula:
[ binom{n}{k} frac{n!}{k!(n-k)!} ]
Where n is the total number of items, and k is the number of items to choose. For our scenario:
Total sample space size [ binom{338}{2} frac{338!}{2!(338-2)!} frac{338 times 337}{2 times 1} 56923 ]
Distinct Jelly Beans
If all the jelly beans were distinct, the number of ways to select two would be given by the combination of 338 choose 2:
[ binom{338}{2} 56923 ]
Non-Distinct Jelly Beans
However, in the case where beans of the same color are indistinguishable, we need to consider the following scenarios:
Case 1: Both Beans of the Same Color
In this case, we choose one of the 6 colors, and the number of ways to select 2 identical jelly beans from a specific color is always 1 (since all beans of the same color are identical). We have 6 such colors, so:
[ 6C1 6 ]
Case 2: Both Beans of Different Colors
For this scenario, we can select two different colors in the following ways:
[ 6C2 frac{6!}{2!(6-2)!} frac{6 times 5}{2 times 1} 15 ]
Therefore, the total number of ways to select two jelly beans of different colors is:
[ 6C1 6C2 6 15 21 ]
Conclusion
By applying the combination formula, we have determined the sample space size for selecting two jelly beans from the jar, considering both distinct and non-distinct scenarios. This combinatorial analysis provides a solid foundation for further probability and statistical studies related to the jar of jelly beans.
Key Terms:
Sample Space: The set of all possible outcomes of an experiment. Jelly Bean Probability: Calculating the likelihood of specific events occurring when selecting jelly beans from a jar. Combination Formula: A mathematical formula used to calculate the number of possible ways to choose a subset of items from a larger set.