Probability of a Yellow Rose Bush Being in the Middle of a Row

A gardener has nine rose bushes to plant: three have red flowers, and six have yellow flowers. To find the probability that a yellow rose bush is in the middle of the row of nine rose bushes, we will use the principles of combinatorial mathematics. This article will provide a step-by-step approach to solving this interesting problem, ensuring it is easily comprehensible for both beginners and advanced learners. By the end, you will understand the detailed derivation and the practical applications of such problems in combinatorics.

Determining Total Arrangements

First, we need to calculate the total number of ways to arrange the 9 rose bushes, given that 3 are red and 6 are yellow. The formula for permutations of items with indistinguishable items is given by:

[text{Total arrangements} frac{n!}{n_1! times n_2!}]

Where n is the total number of items, and n1 and n2 are the counts of the indistinguishable items. Here, n 9, n1 3 (red bushes), and n2 6 (yellow bushes). Let's calculate this step-by-step:

9! 362880
3! 6
6! 720

Substituting these values into the formula:

[text{Total arrangements} frac{362880}{6 times 720} frac{362880}{4320} 84]

Determining Favorable Arrangements

Next, we consider the scenario where a yellow rose bush is placed in the middle of the row, which is the 5th position. This leaves 8 positions for the remaining 5 yellow rose bushes and 3 red rose bushes. We need to calculate the arrangements of these 8 bushes:

[text{Favorable arrangements} frac{8!}{3! times 5!}]

Let's calculate this step-by-step:

8! 40320
3! 6
5! 120

Substituting these values into the formula:

[text{Favorable arrangements} frac{40320}{6 times 120} frac{40320}{720} 56]

Calculating the Probability

Finally, we can calculate the probability that a yellow rose bush is in the middle of the row. This is given by the ratio of favorable arrangements to total arrangements:

[P(text{yellow in middle}) frac{text{Favorable arrangements}}{text{Total arrangements}} frac{56}{84}]

By simplifying this fraction:

[P(text{yellow in middle}) frac{2}{3}]

Thus, the probability that a yellow rose bush is in the middle of the row is .

Explanation

An alternative explanation can be given by considering a different approach. Let's consider the 3 red bushes as a single unit. In this case, we have 7 items to arrange (6 yellow bushes and 1 unit of 3 red bushes). The number of arrangements of these 7 items is 7!. The 3 red bushes can be arranged amongst themselves in 3! ways. Therefore, the numerator becomes 7!3!

Considering all 9 bushes, the total possible arrangements are 9!. Hence:

[P(text{yellow in middle}) frac{7! times 3!}{9!}]

By simplifying this fraction, we arrive at the same result.

Closure: This detailed explanation illustrates how to apply combinatorial mathematics to solve problems related to probability and arrangements. Understanding such concepts is crucial in fields like statistics, data analysis, and combinatorics. Practical applications range from statistical surveys to complex probabilistic models in various scientific and engineering disciplines.