Probability of Randomly Picking Flowers in a Garden

Imagine a garden in the enchanting land of Arden, adorned with a mix of yellow and red roses. To better explore the probabilistic world of this garden, we dive into the joy of random flower plucking. This article delves into the nuances of probability by calculating the chances of plucking specific combinations of flowers, presenting the logic behind the calculations and reinforcing the understanding of combinatorial mathematics.

Understanding the Garden

In the garden of Arden, there are a total of 8 flowers – 5 yellow roses and 3 red roses. Our garden provides the perfect canvas for exploring the fascinating world of probability and combinatorics.

Mathematical Setup

Let's start by calculating the total number of ways to pluck 2 flowers from the garden. We use the binomial coefficient notation to represent combinations.

Total Number of Flower Combinations

The total number of flower combinations can be calculated using the binomial coefficient:

(binom{8}{2} 28)

Calculating Specific Probabilities

Now, let's consider the different scenarios in which we pluck the flowers randomly. We will calculate the probabilities for each of the following specific combinations:

Two Red Flowers Plucked

The number of ways to pluck 2 red flowers from 3 is given by:

(binom{3}{2} 3)

Therefore, the probability of plucking 2 red flowers is:

(frac{3}{28})

Two Yellow Flowers Plucked

Calculating the number of ways to pluck 2 yellow flowers from 5:

(binom{5}{2} 10)

The probability of plucking 2 yellow flowers is:

(frac{10}{28} frac{5}{14})

One Yellow and One Red Flower Plucked

Calculating the number of ways to pluck 1 yellow and 1 red flower:

(binom{5}{1} cdot binom{3}{1} 15)

The probability of plucking 1 yellow and 1 red flower is:

(frac{15}{28})

Verification

To ensure the calculations are correct, we verify the sum of all probabilities:

(frac{3}{28} frac{15}{28} frac{10}{28} frac{28}{28} 1)

This confirms that the total probability is indeed 1, as expected.

Conclusion

Understanding the probabilities in a seemingly simple scenario like picking flowers can be both enlightening and challenging. It provides an excellent example of how combinatorial mathematics can be applied to real-world situations, thereby enhancing our problem-solving skills and logical reasoning abilities.