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Analyze the probability that a randomly selected tulip from a greenhouse with a uniform height distribution between 7 and 16 inches exceeds 10 inches in height.

Given that the heights of tulips follow a continuous uniform distribution with a lower bound of 7 inches and an upper bound of 16 inches, we need to determine the probability that a tulip's height exceeds 10 inches.

The probability density function (PDF) of a uniform distribution is constant over its interval. The total interval length is 16 - 7 = 9 inches. The part of the interval where tulips are tall enough to be selected (height > 10 inches) spans from 10 to 16 inches, which is 6 inches.

Therefore, the probability that a randomly selected tulip exceeds 10 inches in height is the length of this favorable interval divided by the total interval length: P(Height > 10 inches) = (16 - 10) / (16 - 7) = 6 / 9 = 2 / 3 ≈ 0.6667.

Thus, there is approximately a 66.67% chance that a randomly selected tulip from this greenhouse will be tall enough for selection.

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Analysis of Tulip Heights Distribution and Selection Probability

The problem concerns the probability calculation for a tulip's height exceeding a certain threshold, given a uniform distribution of heights. Uniform distributions are characterized by their constant probability density across a specified interval, which simplifies the calculation of probabilities for events within that interval.

In this scenario, the height of tulips in Rotterdam’s Fantastic Flora greenhouse is modeled as a continuous uniform distribution with a lower bound of 7 inches and an upper bound of 16 inches. This implies that each height within this range is equally likely, and the probability density function (PDF) remains constant. The task is to find the probability that a tulip's height exceeds 10 inches, i.e., P(Height > 10 inches).

The total length of the distribution interval is (16 - 7) = 9 inches. The favorable interval where tulips are tall enough (greater than 10 inches) is from 10 inches to 16 inches. The length of this favorable interval is (16 - 10) = 6 inches.

Given the uniform distribution, the probability of a tulip being within a specific interval is proportional to the length of that interval relative to the total. Hence, the probability that a randomly selected tulip exceeds 10 inches in height is:

P(Height > 10 inches) = Length of interval where height > 10 / Total interval length = 6 / 9 = 2 / 3 ≈ 0.6667.

This calculation demonstrates how uniform distributions facilitate straightforward probability computations since the likelihood is directly related to the proportion of the interval lengths.

Environmental factors, cultivation practices, and genetic attributes could influence the actual distribution of tulip heights, but within the simplified uniform distribution model, this probability provides a useful estimate for selection criteria in the greenhouse.

References

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