Probability of Drawing Two Marbles of the Same Color

When dealing with probability problems, particularly those involving marbles, it’s important to break down the problem into simpler, more manageable parts. This article discusses the probability of drawing two marbles of the same color from a bag containing 3 blue and 4 red marbles. We will explain the process step-by-step and provide a final solution.

The Problem

Consider a bag filled with 3 blue and 4 red marbles. A marble is taken at random, its colour is noted, and it is not replaced. Another marble is taken at random. We need to find the probability of drawing two marbles of the same color.

Step-by-Step Solution

We can tackle this problem using the classical probability approach. The classical probability formula is used, which is defined as the number of favorable outcomes divided by the total number of possible outcomes.

Step 1: Total Number of Marbles

The total number of marbles in the bag:

n 3

Step 2: Probability of Drawing Two Blue Marbles

Probability of drawing the first blue marble

After drawing one blue marble, there are 2 blue marbles left and a total of 6 marbles remaining. Therefore, the probability of drawing a second blue marble is

The combined probability of drawing two blue marbles is

Step 3: Probability of Drawing Two Red Marbles

Probability of drawing the first red marble

After drawing one red marble, there are 3 red marbles left and a total of 6 marbles remaining. Therefore, the probability of drawing a second red marble is

The combined probability of drawing two red marbles is

Step 4: Total Probability of Drawing Two Marbles of the Same Color

To find the total probability of drawing two marbles of the same color, we add the probabilities of the two cases.

Final Answer: The probability of drawing two marbles of the same color is 3/7.

Alternative Method and Analysis

An alternative method involves calculating the probability of drawing exactly one blue marble and then the probability of drawing at least one blue marble.

Probability that 1st M is blue and 2nd M is not blue:

Probability that 1st M is not blue and 2nd M is blue:

Therefore, the probability that exactly one blue marble is drawn is:

However, the problem is to find the probability of at least one blue marble. We need to include the probability of both marbles being blue.

Probability that both 1st and 2nd M’s are blue:

Therefore, the probability of at least one blue marble:

This result can also be derived through a simpler function:

While the simpler function is less intuitive, it still provides the correct answer.

Conclusion

In conclusion, we have demonstrated two different methods to solve the problem of finding the probability of drawing two marbles of the same color from a bag containing 3 blue and 4 red marbles. The final probabilities, while presented in different ways, both lead to the conclusion that the probability of drawing two marbles of the same color is (frac{3}{7}) or approximately 0.722 (72.2%).