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A stick is broken at two points, at random.  If the length of the stick is 6, then what is the probability that all three resulting pieces are shorter than 3 units?

 Nov 28, 2022
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A stick is broken at two points, at random.  If the length of the stick is 6, then what is the probability that all three resulting pieces are shorter than 3 units?

 

I'd use contour probability mapping

 

Let one piece be x another  y-x and the third   6-y

 

we know

0

 

y-x>0     which simplifies to    y>x

y-x<6      which simplifies to    y

and

0<6-y<6   which simplifies to  -6<-y< 0   or better still    0

 

Plot all those lines on a number plane.

The area in the middle conatins all the possible outcomes.  

 

Now you want the section inside where  

x<3

y-x<3  and

6-y<3

Find the area contained in this smaller region

 

Put the little area over the big area and you have your probability.

 Nov 28, 2022

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