A geometric probability questionFind the probability of distance of two points ,which are selected in [0,a] closed interval, is less than ka k &lt; 1.

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A geometric probability questionFind the probability of distance of two points ,which are selected in [0,a] closed interval, is less than ka   k  &amp;lt;  1.

Elliott Rollins · Accepted Answer

Step 1The region of       |    y  −  x      |    ≤  k  a looks similar to the picture that I have attached. We just have to use the total area to subtract away the area of the area of the two triangles.Step 2                    a        2            −      2      ⋅              1        2            (      a      −      k      a              )        2                    a      2        =  1  −  (  1  −  k      )    2    =  k  (  2  −  k  )

solvarmedw · Accepted Answer

Step 1We need to find the measure of A, which is given by                               ∫          A                d        x        d        y        =                  ∫          0          a                          ∫                      max            (            y            −            k            a            ,            0            )                                m            i            n            (            y            +            k            a            ,            a            )                          d        x        d        y        =                  ∫          0          a                min        (        y        +        k        a        ,        a        )        −        max        (        y        −        k        a        ,        0        )                d        y        .            .Step 2Note that                               ∫          0          a                min        (        y        +        k        a        ,        a        )                d        y        =                  ∫          0                      a            −            k            a                          y        +        k        a                d        y        +                  ∫                      a            −            k            a                    a                a                d        y        .

Find the probability of distance of two points, which are selected in [0,a] closed interval, is less than ka k<1

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