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: 2

The 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: