The Probability of Picking 4 Green Balls Out of 8: A Detailed Analysis for SEO

In this article, we will delve into the problem of determining the probability of picking 4 green balls out of a set of 8 balls. This problem involves the application of combinatorics and the concept of binomial coefficients. We will provide a step-by-step analysis to help you fully understand the solution and ensure it is optimized for SEO.

Understanding the Problem

The problem at hand is: there are 8 balls in a box3 red balls numbered 1 through 3 and 5 green balls numbered 1 through 5. If you reach into the bag and blindly pick up 4 balls, what is the probability of ending up with 4 green balls?

Step-by-Step Analysis

Step 1: Determine the Total Number of Ways to Choose 4 Balls from 8

First, we need to calculate the total number of ways to choose 4 balls from the 8 available. This can be done using the combination formula:

binom{n}{r} frac{n!}{r!(n-r)!}

Here, n is the total number of items and r is the number of items to choose. In our case, we want to choose 4 balls from 8:

binom{8}{4} frac{8!}{4!8-4!} frac{8 times 7 times 6 times 5}{4 times 3 times 2 times 1} 70

Step 2: Determine the Number of Ways to Choose 4 Green Balls from the 5 Available

Next, we need to find the number of ways to choose 4 green balls from the 5 available green balls. Again, we use the combination formula:

binom{5}{4} frac{5!}{4!5-4!} frac{5}{1} 5

Step 3: Calculate the Probability

The probability of picking 4 green balls is the number of favorable outcomes (choosing 4 green balls) divided by the total outcomes (choosing any 4 balls):

P(4 green balls) frac{binom{5}{4}}{binom{8}{4}} frac{5}{70} frac{1}{14}

Final Answer: The probability of ending up with 4 green balls is 1/14.

SEO Optimized for Google

This article is optimized for search engines by carefully selecting and incorporating relevant keywords, using H tags to improve content structure, providing a detailed analysis, and ensuring the content is as clear and informative as possible. By understanding the probability of picking 4 green balls, readers can apply the same principles to similar combinatorial analysis problems.

Additional Insights

If the balls were not numbered, the analysis would be identical. The number of ways to pick 4 out of 8 distinct balls is given by:

binom{8}{4} frac{8!}{4!4!} 70

And the number of ways to pick 4 green out of 5 green balls is:

binom{5}{4} frac{5!}{4!1!} 5

Thus, the probability remains:

P(4 green balls) frac{binom{5}{4}}{binom{8}{4}} frac{5}{70} frac{1}{14}

If the balls are numbered, the analysis is slightly different, but the probabilities remain the same. The total number of ways to choose 4 distinct balls from 8 is:

binom{8}{4} frac{8!}{4!(8-4)!} 70

And the number of ways to choose 4 distinct green balls from 5 is:

binom{5}{4} frac{5!}{4!1!} 5

Hence, the probability is:

P(4 green balls) frac{5}{70} frac{1}{14}

For this particular situation, the outcome probabilities are identical whether balls are each unique or not.