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. - Decision Point
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.
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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
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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:
