Geometric Probability problem in 3 unknownsSuppose we have to choose 3 numbers, a,b and c such that a , b , c ∈ [ 0 , 1 ]. The numbers are randomply distributed in an uniform distribution between 0 and 1. Then I've been asked to find the probability of a + b &gt; 2 c.I'm not being able to represent this in a geometrical way. I've tried fixing the value of c and then figuring out where a and b would lie on a line segment, but that got me nowhere.How should I approach this particular type of problem?

Question

Geometric Probability problem in 3 unknownsSuppose we have to choose 3 numbers, a,b and c such that   a  ,  b  ,  c  ∈  [  0  ,  1  ]. The numbers are randomply distributed in an uniform distribution between 0 and 1. Then I&#039;ve been asked to find the probability of   a  +  b  &amp;gt;  2  c.I&#039;m not being able to represent this in a geometrical way. I&#039;ve tried fixing the value of c and then figuring out where a and b would lie on a line segment, but that got me nowhere.How should I approach this particular type of problem?

Cynthia George · Accepted Answer

Explanation:                              P                (        a        +        b        &amp;gt;        2        c        )                            =                  E                [                  P                (        a        +        b        &amp;gt;        2        c        ∣        c        )        ]        =                  ∫          0          1                          P                (        a        +        b        &amp;gt;        2        x        )                d        x                                          =                  ∫          0                      1                          /                        2                                    (          1          −          2                      x            2                    )                        d        x        +                  ∫                      1                          /                        2                    1                2                              (            1            −            x            )                    2                        d        x        =                  1          2                .

Bobby Mitchell · Accepted Answer

Step 1You could think of a,b,c are the three coordinates of a point in   1  ×  1  ×  1 cube, then you want to find the volume of portion of the cube where the z coordinate c is less than half the sum of the x and y coordinates.Step 2The plane separating the two disjoint regions is   z  =            1      2        (  x  +  y  ).Therefore,   P  =  Volume  =            ∫              x        =        0                    1                    ∫              y        =        0                    1                            1        2              (    x    +    y    )    d    y    d    x    =                  1        2                    ∫              x        =        0            1        (    x    +                  1        2              )    d    x    =                  1        2              (                  1        2              +                  1        2              )    =                  1        2

Suppose we have to choose 3 numbers, a, b and c such that a, b, c in [0,1]. The numbers are randomply distributed in an uniform distribution between 0 and 1. Then I've been asked to find the probability of a+b>2c.

Open question

Answer & Explanation

New Questions in High school geometry