Probability of Drawing Specific Marbles from an Urn
In this article, we will explore the concept of probability, specifically focusing on the scenario of drawing specific marbles from an urn. We'll delve into the mathematics behind combinations, and discuss the differences between drawing with and without replacement. The content is structured to cater to individuals interested in statistics, probability theory, and combinatorics.
Introduction to the Problem
The scenario involves an urn containing 8 red, 6 blue, 4 green, and 5 yellow marbles. Our objective is to calculate the probability of drawing two blue marbles and one yellow marble in a random selection of three marbles. This problem provides an excellent opportunity to apply fundamental principles of probability and combinations.
Understanding the Setup
The urn contains a total of 8 red, 6 blue, 4 green, and 5 yellow marbles, summing up to 23 marbles. We are interested in calculating the probability of drawing two blue and one yellow marble when three marbles are picked up at random. The key steps in solving this problem include determining the total number of ways to choose three marbles from the urn and calculating the favorable outcomes.
Calculating the Total Number of Ways to Choose 3 Marbles from 23
Using the combination formula, binom{n}{r} frac{n!}{r!(n-r)!}, we can determine the total number of ways to choose 3 marbles from 23. The calculation is as follows:
[text{Total ways} binom{23}{3} frac{23 times 22 times 21}{3 times 2 times 1} 1771]
Calculating Favorable Outcomes
To find the favorable outcomes, we need to determine the number of ways to choose 2 blue marbles from the 6 blue marbles and 1 yellow marble from the 5 yellow marbles. Using the combination formula again, we have:
[text{Ways to choose 2 blue marbles} binom{6}{2} frac{6 times 5}{2 times 1} 15]
[text{Ways to choose 1 yellow marble} binom{5}{1} 5]
[text{Favorable outcomes} text{Ways to choose 2 blue marbles} times text{Ways to choose 1 yellow marble} 15 times 5 75]
Calculating the Probability
The probability is then calculated as the ratio of the favorable outcomes to the total number of ways to choose 3 marbles. Thus:
[text{Probability} frac{75}{1771}]
The final probability of drawing 2 blue marbles and 1 yellow marble is:
[boxed{frac{75}{1771}}]
Considering Different Sampling Methods
It is also interesting to consider how the probability changes if the marbles are drawn with or without replacement.
Without Replacement:
The probability can be calculated as:
[binom{6}{2} times binom{5}{1} / binom{23}{3} frac{15 times 5}{1771} frac{75}{1771}]
This is consistent with the initial calculation.
With Replacement:
In this case, the composition of the urn remains the same before each draw. The probability can be calculated as:
[left(frac{6}{23}right) times left(frac{6}{23}right) times left(frac{5}{23}right) times 3! / 2! frac{540}{12167}]
or approximately:
[0.0423]
Conclusion
This article has explored the probability of drawing specific marbles from an urn with a detailed explanation of the calculations. Understanding the difference in probabilities when drawing with or without replacement provides insight into the nuances of probability theory. This knowledge is invaluable for anyone interested in statistics, combinatorics, and probability theory.