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# Hard Problem

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A line segment of length 5 is broken at two random points along its length. What is the probability that the shortest of the three new segments has length longer than 1?

Guest Jul 13, 2018
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I would use probablility contour mapping to answer this question.

A line segment of length 5 is broken at two random points along its length. What is the probability that the shortest of the three new segments has length longer than 1?

Let the lengths be x,   5-y and  y-x

Note that when you add all these you get a lenght of 5

so

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Lets graph that.

The darkest area represents the entire sample space.   How much area does it have?

Area = 0.5*5*5 = 12.5 units squared

Now how much of this sample space will have the shortest side bigger than than 1 unit?

mm

I think it is a lot easier to graph the sample space where one side is less than 1.

These are mutually exclusive and exhaustive events so their probabilities will be add up to 1

The lengths are  x,   5-y and  y-x

When will a length be LESS than 1 unit

$$0 The dark shaded area is where there IS a side that is less then one unit. SO the triangle in the middle is where there is NO SIDE less than 1 unit. Area of trianlge in middle = 0.5*2*2 = 2 units squared. So the probability that the shortest length will be longer than 1 unit is \(\frac{2}{12.5}= \frac{4}{25}=\frac{16}{100}=16\%$$

Melody  Jul 15, 2018
edited by Melody  Jul 15, 2018