Probability of Drawing Balls of the Same Color from Two Bags

Imagine you have two bags, each filled with a unique combination of colored balls. Bag 1 contains 6 red, 5 blue, and 4 green balls, while Bag 2 contains 5 red, 6 green, and 4 blue balls. What is the probability that if you draw one ball from each bag, both balls will be of the same color?

Setting Up the Problem

To solve this problem, we need to calculate the individual probabilities of drawing each color from each bag, and then combine these probabilities to find the overall probability of drawing two balls of the same color. Let's start by calculating the probability of drawing each color from each bag.

Bag 1

Red balls: 6 out of 15, so the probability (PR/Bag1) is 6/15 Blue balls: 5 out of 15, so the probability (PB/Bag1) is 5/15 Green balls: 4 out of 15, so the probability (PG/Bag1) is 4/15

Bag 2

Red balls: 5 out of 15, so the probability (PR/Bag2) is 5/15 Blue balls: 4 out of 15, so the probability (PB/Bag2) is 4/15 Green balls: 6 out of 15, so the probability (PG/Bag2) is 6/15

Notice that there was a correction needed in the description of Bag 2. It should have 4 blue balls instead of 4 red balls.

Calculating the Combined Probability

To find the probability of drawing two balls of the same color, we multiply the probabilities of drawing each color from both bags. We will do this for each color and then sum the probabilities.

Same Red Balls

The probability of drawing a red ball from both bags is:

(PR/Bag1) * (PR/Bag2) (6/15) * (5/15) 30/225

Same Blue Balls

The probability of drawing a blue ball from both bags is:

(PB/Bag1) * (PB/Bag2) (5/15) * (4/15) 20/225

Same Green Balls

The probability of drawing a green ball from both bags is:

(PG/Bag1) * (PG/Bag2) (4/15) * (6/15) 24/225

Now, to find the total probability of drawing two balls of the same color, we add these probabilities together:

(30/225) (20/225) (24/225) (74/225)

Converting 74/225 to a decimal gives us:

0.328889

Therefore, the probability of drawing two balls of the same color from these two bags is 0.328889, or approximately 32.89%.

Conclusion

By carefully calculating the individual probabilities and combining them, we have determined that the likelihood of drawing two balls of the same color from Bag 1 and Bag 2 is 32.89%. This method can be applied to similar problems in probability, providing a logical and structured approach to solving such questions.

Understanding and applying these principles of probability can be tremendously useful in various fields, from data analysis to statistical modeling. Whether you are a student, a researcher, or an enthusiast of mathematics, mastering these concepts can greatly enhance your analytical skills.

Key Takeaways:

The probability of drawing a ball of a specific color from a bag is calculated by the ratio of the number of balls of that color to the total number of balls in the bag. To find the probability of two independent events happening together (in this case, drawing two balls of the same color), multiply the individual probabilities of each event. Applying these principles can help in solving a wide range of probability-related problems.

For further practice, consider solving more complex probability problems, and always check your work to ensure accuracy in your calculations.