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. - RoadRUNNER Motorcycle Touring & Travel Magazine
Title: Probability of Drawing Exactly 2 Red and 2 Blue Marbles from a Bag âÃÂàStep-by-Step Calculation
Title: Probability of Drawing Exactly 2 Red and 2 Blue Marbles from a Bag âÃÂàStep-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 âÃÂà7 red and 9 blue âÃÂàusing 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 doesnâÃÂÃÂt 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:
📰 Trails of Cold Steel Walkthrough 📰 Okami Walkthrough Ps3 📰 Pokemon Black 2 Map 📰 Hormita Revealed The Hidden Fat Burning Secret Everyones Craving No More Guesswork 8723396 📰 Stock Market News November 4 2025 9214611 📰 Wall E Game 📰 Call Epic Games Phone Number 1788423 📰 Market Basket Employee Suspension 7850514 📰 Find My Phone Iphone 📰 Crack The Truth Stock Markets Shut Down On Presidents Dayheres Why 9884661 📰 Bank Of America Golf 📰 Fda Black Box Hormone Therapy 📰 Program Guide Process Lasso Download Direct Source 📰 Finally What Is Ssms In Sql Server The Shorts Answer That Will Change Your Query Game 5792674 📰 Red The Movie 1641714 📰 Car Insurance Ny Cheap 📰 Plumbers Fredericksburg Va 4092251 📰 Best Android Phone 2024Final 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:
