Two points are chosen at random on a line segment of length 9 cm. The probability that the distance between these 2 points is less than 3 cm is?

Question

kidoceanoe · Accepted Answer

Step 1Here is how to start the problem:Let       X    1    ,      X    2    ∼  Uni  ⁡  (  0  ,  9  )      X    i   takes value from 0 to 9, the total area would be 92.Step 2We have to find the area of the region that corresponds to       |        X    1    −      X    2        |    &amp;lt;  3 then divide by the total area.

Mark Elliott · Accepted Answer

Step 1On Careful Observation we can spot two parallelograms each of equal area and two triangles each of equal area which makes the calculation of shaded area a very easy task. The first parallelogram has coordinates at (0,0),(0,3),(6,9),(6,6) giving area   =  3  ⋅  6  =  18 units. Similarly another parallelogram with coordinates at (3,0), (3,3),(9,9),(9,6) has area of   3  ⋅  6   (  base  ⋅  height  )  =  18.Step 2Now remaining two triangles have same dimensions base   =  3  and height   =  3. Area   =  4.5 each. Summing to   18  +  18  +  9  =  45.

Two points are chosen at random on a line segment of length 9 cm. The probability that the distance between these 2 points is less than 3 cm is?

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