Conditional Probability of Drawing a White Ball from a Box
Conditional Probability of Drawing a White Ball from a Box
Understanding the principles of conditional probability can be crucial in various real-world scenarios, from statistical analysis to lottery odds prediction. In this article, we will delve into the concept of conditional probability, focusing on an example where we calculate the probability of drawing a white ball as the second ball, given that the first ball drawn is white. This article is designed to be easily digestible and SEO-friendly, ensuring it meets Google's standards for quality content.
Initial Setup and Problem Statement
Consider a box containing 7 red balls and 5 white balls, making a total of 12 balls. The problem at hand is to find the probability that the second ball drawn is white given that the first ball drawn is also white.
Step-by-Step Solution
Initial Setup: The box contains 7 red balls and 5 white balls, resulting in a total of 12 balls.
First Draw: If the first ball drawn is white, then:
4 white balls remain, since one white ball has been drawn. 7 red balls remain. Total remaining balls 4 (white) 7 (red) 11 balls.Second Draw: We want to find the probability that the second ball drawn is white, given that the first ball drawn was white. The number of favorable outcomes (drawing a white ball) is 4, while the total number of possible outcomes is 11.
The probability is calculated as follows:
[ P(text{Second ball is white} mid text{First ball is white}) frac{text{Number of remaining white balls}}{text{Total remaining balls}} frac{4}{11} ]
Hence, the probability that the second ball is white given that the first ball is white is
[frac{4}{11}].
Alternative Approaches for Understanding
Several alternative methods can be used to understand the problem and validate the result. For instance, consider the following approaches:
Sample Space and Combinations
The first draw can be either red or white. If the first ball is red or white, the probabilities for the second draw are as follows:
Probability of red first and white second: ( frac{7}{12} times frac{5}{11} frac{35}{132} ) Probability of white first and white second: ( frac{5}{12} times frac{4}{11} frac{20}{132} )The total probability is the sum of these two probabilities:
[ frac{35}{132} frac{20}{132} frac{55}{132} frac{5}{12} ]
However, we are interested only in the cases where the second ball is white, so we need to divide by the total probability of drawing a white ball as the second:
[ frac{frac{20}{132}}{frac{55}{132}} frac{20}{55} frac{4}{11} ]
Equal Distribution
Considering that any of the 5 remaining white balls has an equal chance of being drawn as the second ball, and 2 out of these 5 balls are white, the probability of drawing a white ball as the second is:
[ frac{2}{5} ]
This is a simplified approach that can help in understanding the concept without delving into detailed combinatorial analysis.
Conclusion
Understanding conditional probability is vital for many applications in statistics, finance, and decision-making. The probability that the second ball drawn is white, given that the first ball drawn is white, is (frac{4}{11}). This article provides a clear and step-by-step explanation, along with alternative methods, to ensure a comprehensive understanding of the concept.