Question: A box contains 7 red marbles and 9 blue marbles. If four marbles are drawn without replacement, what is the probability that exactly two are red? - AIKO, infinite ways to autonomy.
Understanding the Probability of Drawing Exactly Two Red Marbles from a Box (7 Red & 9 Blue)
Understanding the Probability of Drawing Exactly Two Red Marbles from a Box (7 Red & 9 Blue)
When faced with a probabilistic scenario like drawing marbles from a bag, understanding the underlying math not only satisfies curiosity but also strengthens critical thinking. A particularly common and clear example involves calculating the likelihood of selecting exactly two red marbles out of four drawn without replacement from a box containing 7 red and 9 blue marbles. In this article, weΓΒΓΒ’ΓΒΓΒΓΒΓΒll break down the problem step by step to explain how to calculate this probability and why it matters.
Understanding the Context
The Problem: Drawing 4 Marbles Without Replacement
We are given:
- 7 red marbles
- 9 blue marbles
- Total marbles = 7 + 9 = 16
- Number of marbles drawn = 4 (without replacement)
- We want exactly 2 red and 2 blue marbles
Image Gallery
Key Insights
Why Use Combinations?
Since the marbles are drawn without replacement, each draw affects the composition of the remaining marbles. This is a hypergeometric distribution scenarioΓΒΓΒ’ΓΒΓΒΓΒΓΒideal for calculating probabilities involving finite populations with two outcomes (here, red or blue).
Instead of calculating probabilities for each individual order (like R-R-B-B), we compute all favorable outcomes over total possible outcomes.
Step-by-Step Calculation
π Related Articles You Might Like:
π° Dont Believe These Shocking 401(k) Averages by Age β Your Future Depends on This! π° How Much Is Your 401(k)? The Eye-Opening Average Balance by Age Everyone Should Know! π° 401(k) Balance Breakdown by Age: Discover the Surprising Truth You Cant Ignore! π° Nightwing Teen Titans 9335948 π° The Hidden Genius Behind This Overflowing Nerd Project Left Everyone Askingwhere Did It All Go 5327420 π° Learn How Solitaire Word Games Outperform Other Casual Games In 2024 2920667 π° Locate My Apple 4063421 π° The Secret Pocket Minecraft Trick Thats Taking The Game By Storm 1965181 π° Get The Ultimate Nintendo Switch Case Protection You Cant Ignore 931340 π° The Last Of Us Ellie Revealed The Final Chapter No One Was Prepared For 6460765 π° Unlock The Ultimate Pokmon Monster Powerplay The Hottest Game Of The Year 5781000 π° Gjetost Cheese 3063972 π° Installer Apk Windows 9011587 π° Wait Perhaps The Researcher Uses Expected Value So Fractional Is Acceptable In Calculation But Final Answer Should Be Integer 7600529 π° Number To The Zero Power 7652141 π° Wiseview Exposed How This Smart Platform Transforms Everyday Choices 8807804 π° Pro Tips From Top Oracle Sql Developers Thatll Transform Your Database Workflow 4352428 π° Crazy Flips 3D Unblocked Watch These Impossible Tricks Go Insane 9551085Final Thoughts
Step 1: Determine total ways to draw 4 marbles from 16
This is the total number of possible combinations:
[
inom{16}{4} = rac{16!}{4!(16-4)!} = rac{16 \ imes 15 \ imes 14 \ imes 13}{4 \ imes 3 \ imes 2 \ imes 1} = 1820
]
Step 2: Count the favorable outcomes (exactly 2 red 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
]
Multiply these to get all favorable combinations:
[
21 \ imes 36 = 756
]
Step 3: Compute the probability
[
P(\ ext{exactly 2 red}) = rac{\ ext{favorable outcomes}}{\ ext{total outcomes}} = rac{756}{1820}
]
Simplify the fraction:
Divide numerator and denominator by 4:
[
rac{756 \div 4}{1820 \div 4} = rac{189}{455}
]
This fraction is in simplest form, though decimal approximation is useful:
