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. - Coaching Toolbox
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:
📰 sextpanther 📰 sexual and sexual reproduction 📰 sexxy red real name 📰 The Girl From Arkanya 1950810 📰 The Shocking Truth Behind P0011 Explodes Sensors All Over The World 3966276 📰 From Royalty To Rebellion The Maud Pie Mystery That Users Are Obsessed With 5121750 📰 You Wont Believe The Hidden Pay Of A Private Trainer Job 7876553 📰 Herfindahl Hirschman Index 1121580 📰 The Rockford Fosgate Hype Youve Been Ignoringheres Why Everyone Talks About It 6089118 📰 Russia Google 254533 📰 A Haunting In Venice 5301526 📰 My Verizon Pay Bill Online 8992067 📰 Earth Destroy 9381310 📰 Timeline That Shocks Golden State And Orlando Magic Fight That Will Renter Legends Forever 9215755 📰 Party City Head Office 2262007 📰 Why This Unique B Player Name Stands Out In Every Basketball Court 9899086 📰 Pressure In Barometer 3236662 📰 Abducens Nerve 1919763Final 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:
