A song's length in a short window

Question

A streaming service generates jingles whose lengths are equally likely to be anything between 3.0 and 5.0 minutes. So the length is uniform across a 2.0-minute range.

What is the probability that a jingle lasts between 3.4 and 3.8 minutes?

Accepted Answer

For a length spread evenly over an interval, the probability of landing in a target slice is that slice's width divided by the width of the whole range. P(	ext{event}) = \frac{	ext{width of favorable slice}}{	ext{width of full range}}. The full range is 5.0 - 3.0 = 2.0 minutes wide. The favorable slice, from 3.4 to 3.8 minutes, is 3.8 - 3.4 = 0.4 minutes wide. So P(3.4 \le 	ext{length} \le 3.8) = \frac{0.4}{2.0} = \frac{1}{5} = 0.2. The slice is narrow (0.4 min) compared with the 2.0-minute range, so it is no surprise the probability is small. Takeaway. A uniform value's probability over …

A song's length in a short window

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