Solution: We calculate the probability of drawing exactly 2 red marbles and 2 blue marbles from a total of 16 marbles (7 red, 9 blue), without replacement. - Databee Business Systems
Title: Probability of Drawing Exactly 2 Red and 2 Blue Marbles from a Bag â Step-by-Step Calculation
Title: Probability of Drawing Exactly 2 Red and 2 Blue Marbles from a Bag â Step-by-Step Calculation
When sampling without replacement, understanding the likelihood of mixed outcomes becomes essential in probability and statistics. In this article, we explore how to calculate the probability of drawing exactly 2 red marbles and 2 blue marbles from a total of 16 marbles â 7 red and 9 blue â using combinatorial methods.
Understanding the Context
Problem Statement
We have a bag containing:
- 7 red marbles
- 9 blue marbles
Total = 16 marbles
We draw 4 marbles without replacement and want to compute the probability of obtaining exactly 2 red and 2 blue marbles.
Image Gallery
Key Insights
Why Use Combinations?
Because the drawing is without replacement and order doesnâÂÂt matter, we use combinations to count possible selections. The probability is calculated as:
[
P(2, \ ext{red and } 2, \ ext{blue}) = rac{\ ext{Number of favorable outcomes}}{\ ext{Total number of possible 4-marble combinations}}
]
Step 1: Calculate Total Possible Outcomes
🔗 Related Articles You Might Like:
North & Northwest Movie: Breakinglatest Secrets That Will Blow Your Mind! Discover the Hidden Gems in the North & Northwest Movie—You’ll Be Addicted! 🔥 Discover the Secret Power of the North Arrow – What It Reveals About Your Navigation!Final Thoughts
The total number of ways to choose any 4 marbles from 16 is given by the combination formula:
[
inom{n}{k} = rac{n!}{k!(n-k)!}
]
So,
[
inom{16}{4} = rac{16!}{4! \cdot 12!} = 1820
]
Step 2: Calculate Favorable Outcomes
We want exactly 2 red marbles and 2 blue marbles:
-
Ways to choose 2 red marbles from 7:
[
inom{7}{2} = rac{7 \ imes 6}{2 \ imes 1} = 21
] -
Ways to choose 2 blue marbles from 9:
[
inom{9}{2} = rac{9 \ imes 8}{2 \ imes 1} = 36
]
Since the selections are independent, multiply the combinations:
