Probability of Drawing Two Marbles of the Same Color from a Bag
In this article, we will explore the probability of drawing two marbles of the same color from a bag containing 5 blue, 3 green, and 2 red marbles. We will use combinatorial mathematics to arrive at the solution and present a step-by-step approach.
Introduction to the Problem
A bag contains 5 blue marbles, 3 green marbles, and 2 red marbles. If two marbles are drawn without replacement, what is the probability that they will be the same color?
Total Number of Marbles and Ways to Draw Two
First, let’s calculate the total number of marbles in the bag:
Total marbles 5 blue 3 green 2 red 10 marbles
To find the total number of ways to draw two marbles from the bag, we use the combination formula, which is given by:
C(n, k) n! / (k! * (n - k)!)
For our case, where n 10 and k 2:
C(10, 2) 10! / (2! * (10 - 2)!)
Calculating this, we get:
C(10, 2) 45
Ways to Draw Two Marbles of the Same Color
Now, let’s calculate the number of ways to draw two marbles of the same color for each color:
Blue Marbles
C(5, 2) 5! / (2! * (5 - 2)!) 10
Green Marbles
C(3, 2) 3! / (2! * (3 - 2)!) 3
Red Marbles
C(2, 2) 2! / (2! * (2 - 2)!) 1
Adding these together gives us the total number of ways to draw two marbles of the same color:
10 (blue) 3 (green) 1 (red) 14
Probability of Drawing Two Marbles of the Same Color
The probability that the two marbles drawn are of the same color is then calculated by dividing the number of favorable outcomes by the total outcomes:
P(same color) 14 / 45
Thus, the probability that the two marbles drawn will be of the same color is:
boxed{14/45}
Conclusion
Using the principles of combinatorial mathematics, we have determined that the probability of drawing two marbles of the same color from a bag containing 5 blue, 3 green, and 2 red marbles is 14/45 or approximately 31.11%.