I set up the sample space as a 10x10 square where each axis is the start time of one flower, then figured out which region of that square leads to overlap.
Model the start times as independent uniform random variables on [0,10]. The flowers overlap if the absolute difference between their start times is less than the sum of their bloom durations (4+2=6) and also the earlier flower hasn't finished before the later starts. Compute the area in the 10x10 square where this condition holds, then divide by 100.
Pro tip: Draw a diagram of the sample space and shade the overlap region. This visual approach makes the probability calculation intuitive and helps avoid off-by-one errors in the inequality.
Let X be the purple flower's start time and Y be the red flower's start time, both uniformly distributed on [0,10] and independent.
The flowers overlap if the intervals [X, X+4] and [Y, Y+2] intersect. This occurs when X ≤ Y+2 and Y ≤ X+4, i.e., |X - Y| ≤ 4? Wait, careful: The condition is X ≤ Y+2 and Y ≤ X+4, which simplifies to -4 ≤ X - Y ≤ 2. So the difference D = X - Y must be between -4 and 2.
In the 10x10 square of possible (X,Y) pairs, the overlap region is the set where -4 ≤ X - Y ≤ 2. Calculate the area of this region by integrating or using geometric shapes, then divide by 100.
The area can be found by subtracting the areas of two triangles where the condition fails: one where X - Y > 2 and one where X - Y < -4. Each triangle has legs of length 8 and 6 respectively? Actually, for X - Y > 2, the region is a triangle with vertices (2,0), (10,0), (10,8) – area = 0.5*8*8 = 32. For X - Y < -4, the region is a triangle with vertices (0,4), (0,10), (6,10) – area = 0.5*6*6 = 18. Total non-overlap area = 50, so overlap area = 100 - 50 = 50. Probability = 0.5.
Double-check the inequalities and area calculations. Present the final probability as 1/2 or 50%.
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