Probability of Drawing Specific Marbles from a Jar
Imagine a jar that contains a total of 14 marbles: 3 red, 5 blue, and 6 white. Each marble is uniquely labeled, and you're interested in the probability of drawing a specific combination of marbles: 2 red, 1 blue, and 1 white, in any order. This article will break down the process of calculating this probability and provide a deeper understanding of the underlying mathematics.
Understanding the Probability Without Replacement
When drawing marbles without replacement, the probability of drawing a specific marble changes with each draw. The initial probability of drawing a red marble is 3/14, a blue marble is 5/14, and a white marble is 6/14. If we focus on the sequence "red, red, blue, white," the probability of drawing these marbles in that exact order would be:
[frac{3}{14} times frac{3}{13} times frac{5}{12} times frac{6}{11} frac{135}{19208} approx 0.007]
However, since the sequence doesn't matter, we need to account for all possible orders. There are (frac{4!}{2!1!1!} 12) different ways to draw 2 red, 1 blue, and 1 white marble. Therefore, the total probability is:
[12 times frac{135}{19208} frac{1620}{19208} frac{405}{4802} approx 0.084]
Mathematical Explanation and Combinatorial Approach
Alternatively, we can use combinatorial methods to determine the probability. The total number of ways to draw 4 marbles out of 14 is given by the binomial coefficient:
[binom{14}{4} frac{14!}{4!(14-4)!} 1001]
The number of ways to choose 2 red marbles out of 3, 1 blue marble out of 5, and 1 white marble out of 6 is:
[binom{3}{2} times binom{5}{1} times binom{6}{1} 3 times 5 times 6 90]
Therefore, the probability of drawing 2 red, 1 blue, and 1 white marble is:
[frac{90}{1001} approx 0.0899]
Conclusion
This problem demonstrates the importance of understanding both the need to account for different orders and the use of combinatorial methods in calculating probabilities. Whether you calculate it through the specific sequence approach or the combinatorial method, the probability for the desired outcome remains a valuable lesson in probability theory and combinatorics. Understanding these concepts can be crucial in various fields, from statistical analysis to game theory.
Frequently Asked Questions
1. Can I use a different sequence for the calculation? Yes, you can use any sequence, but you need to account for all possible orders, as in the combinatorial method.
2. Is it easier to calculate the probability with replacement? No, without replacement, the probability for each subsequent draw changes, making the calculations more complex.
3. Why is labeling irrelevant in the combinatorial method? Labeling is irrelevant because we are calculating the number of combinations, not permutations, and the order in which we pick the marbles does not matter.