Probability of Drawing Blue Marbles from a Bag
Understanding probability, especially in scenarios involving combinations, is crucial in various fields including gambling, statistics, and even daily decision-making. In this article, we will explore the probability of drawing blue marbles from a bag containing a mix of red, blue, and green marbles.
Introduction to the Problem
A bag contains 8 red marbles, 3 blue marbles, and 2 green marbles. We want to calculate the exact probability of drawing two marbles of the same color, specifically both blue. We will employ the concepts of combinations and probability to arrive at a precise answer.
Step-by-Step Calculation
Step 1: Determine the Total Number of Marbles
First, let's determine the total number of marbles in the bag:
Red: 8 Blue: 3 Green: 2The total number of marbles is:
8 3 2 13 marbles.Step 2: Calculate the Total Number of Ways to Draw 2 Marbles
To choose 2 marbles out of 13, we use the combination formula:
(binom{n}{r} frac{n!}{r!(n-r)!})
For our case:
(binom{13}{2} frac{13!}{2!(13-2)!} frac{13 times 12}{2 times 1} 78)
Step 3: Calculate the Number of Favorable Outcomes Drawing 2 Blue Marbles
To choose 2 blue marbles out of 3, we use the combination formula:
(binom{3}{2} frac{3!}{2!(3-2)!} frac{3 times 2}{2 times 1} 3)
Step 4: Calculate the Probability
Now, we can calculate the probability that both marbles drawn are blue:
(P(text{both blue}) frac{text{Favorable outcomes}}{text{Total outcomes}} frac{3}{78})
To simplify (frac{3}{78}), we get:
(frac{3}{78} frac{1}{26})
Step 5: Final Answer
The exact probability that both marbles drawn will be blue is:
boxed{frac{1}{26}}
Probability of Drawing a Blue Marble
Let's also calculate the probability of drawing a single blue marble from the bag. The total number of marbles in the bag is 15 (8 red 3 blue 2 green).
Using the formula:
(text{Probability} frac{text{Number of Favorable Outcomes}}{text{Total Number of Possible Outcomes}})
The number of favorable outcomes is the number of blue marbles (4) and the total number of possible outcomes is the total number of marbles in the bag (15).
(text{Probability} frac{4}{15})
Simplifying this fraction, we get:
(frac{1}{3})
Thus, the probability of drawing a blue marble from the bag is:
(frac{1}{3})
Conclusion
This article has provided a detailed and step-by-step approach to understanding and calculating probability, specifically in scenarios involving combinations. Whether you are a student, a professional, or simply someone interested in probability, the knowledge and techniques explored here can be incredibly useful.