Probability of Drawing Two Green Balls from a Jar without Replacement

In this article, we will explore how to calculate the probability of drawing two green balls from a jar without replacement. This involves understanding the principles of probability and combinations, which are fundamental concepts in statistics and have wide applications in various fields.

Introduction

A jar contains 4 red balls, 6 blue balls, and 8 green balls. We want to determine the probability of drawing two green balls in succession without replacement. In other words, once a ball is drawn, it is not returned to the jar, and the second draw is made with the remaining balls.

Step-by-Step Calculation

Initial Setup:
Let's denote the total number of balls in the jar as N 18. First Draw:
The probability of drawing a green ball on the first draw is given by the ratio of the number of green balls to the total number of balls. Therefore, we have: Second Draw:
After drawing a green ball on the first draw, there are now 17 balls left in the jar, with 7 of them being green. So, the probability of drawing a second green ball is: Total Probability:
The probability of both events happening (i.e., drawing two green balls in succession) is the product of the individual probabilities:

Mathematical Calculation

Let's perform the mathematical calculation step-by-step:

The probability of drawing the first green ball is 8/18. After drawing the first green ball, there are now 7 green balls left out of a total of 17 balls. So, the probability of drawing the second green ball is 7/17. Therefore, the probability of both balls being green is:

8/18 * 7/17 56/306, which simplifies to 28/153.

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Probability of Drawing Balls of the Same Color

Let's explore the probability of drawing two balls of the same color from the jar:

Red Balls:
The probability that the first ball is red is 4/18. After drawing a red ball, the probability of drawing a second red ball from the remaining 17 balls is 3/17. Therefore, the probability of drawing two red balls is: Blue Balls:
The probability that the first ball is blue is 6/18. After drawing a blue ball, the probability of drawing a second blue ball from the remaining 17 balls is 5/17. Therefore, the probability of drawing two blue balls is: Green Balls:
Since there is only one green ball, it is impossible to draw two green balls from the jar. The probability is 0.

Calculation

The probability that the two balls are the same color is the sum of the probabilities of drawing two red balls, two blue balls, and two green balls:

3/14 3/28 - 9/28 6/28 3/28 - 9/28 0.

Combination Approach

Another way to calculate the probability is to use the concept of combinations:

We have 8 green balls, and we want to choose 2 out of these 8. The number of ways to do this is given by the combination formula: {^8C_2} 8! / (2! * 6!) 28. The total number of ways to choose 2 balls out of 18 is {^18C_2} 18! / (2! * 16!) 153.

Therefore, the probability of drawing two green balls is:

{28}/{153}.

So, the probability that both balls drawn are green is {28}/{153}.

Conclusion

This article has provided a comprehensive look at calculating the probability of drawing two green balls from a jar with 4 red, 6 blue, and 8 green balls without replacement. We used both direct probability calculation and the combination approach to arrive at the final answer.

Further Reading

For those interested in deeper explorations of probability and combinatorics, consider checking out these related articles:

Basic Concepts of Probability Combinatorics and Probability Probabilistic Reasoning in Everyday Life