Probability of Drawing Same-Colored Balls from a Bowl
Introduction
This article discusses the probability of drawing two balls of the same color from a bowl containing a certain number of red and black balls. By understanding the principles of probability and applying combinatorial methods, we can calculate the likelihood of drawing either two red or two black balls random drawing. Let's explore this concept in detail.
Understanding the Problem
Consider a bowl containing 5 red balls and 6 black balls. We randomly pick two balls simultaneously. What is the probability that both balls are of the same color?
The probability of drawing two balls of the same color can be calculated as the sum of the probabilities of drawing two red balls and two black balls.
Calculating the Probability
P(BOTH same color) P(both red) P(both black)
P(both red) 6C2 / 11C2
P(both black) 6C2 / 11C2
The notation nCr represents the number of combinations of n items taken r at a time, which is calculated using the formula nCr n! / [r!(n-r)!].
P(both red) 6! / [2!(6-2)!] / 11! / [2!(11-2)!] 15/55 3/11
P(both black) 6! / [2!(6-2)!] / 11! / [2!(11-2)!] 15/55 3/11
P(both same color) 3/11 3/11 6/11
P(both different color) 1 - P(both same color) 1 - 6/11 5/11
Another Approach: Conditional Probability
We can also solve this problem using the conditional probability method.
P(red, red) (6/11) * (5/10) 30/110
P(black, black) (5/11) * (4/10) 20/110
P(both same color) P(red, red) P(black, black) 30/110 20/110 50/110 5/11
The probability that both balls are of the same color is thus 5/11, or approximately 45.45%.
Conclusion
This problem illustrates the application of probability theory in real-world scenarios. By understanding the principles of combinatorial methods and conditional probability, we can solve such problems accurately. Whether you are interested in statistics, mathematics, or simply enjoy a challenge, exploring probability problems like this one can be both fun and educational.
Related Keywords
Probability, same colored balls, random drawing