What is this geometric probabilityIn a circle of radius R two points are chosen at random(the points can be anywhere, either within the circle or on the boundary). For a fixed number c, lying between 0 and R, what is the probability that the distance between the two points will not exceed c?

Question

phinny5608tt · Accepted Answer

Step 1Basically you have to distinguish two cases: A circle with radius c around the first point lies completely inside the circle that is span by R. Then the probability is             c      2              R      2      , but this only happens in             (      R      −      c              )        2                    R      2       of the cases.Step 2Therefore the total probability will be                     c        2            ⋅      (      R      −      c              )        2                    R      2        +  K  ⋅      (    1    −                  (        R        −        c                  )          2                            R        2              )  .With K being the probability of the distance between the two points larger then c in case of the circle around the first point with radius c intersecting with the outter circle. And       (    1    −                  (        R        −        c                  )          2                            R        2              )   being the probability for this to happen.In order to calculate K you need the formula for circle-circle intersections:Step 2Given two circles with radi R and r the formula for the intersectionarea A is:  A  (  R  ,  r  ,  d  )  =      r    2        cos          −      1        ⁡      (                            d          2                +                  r          2                −                  R          2                            2        d        r              +          R      2        )        cos          −      1        ⁡      (                            d          2                +                  R          2                −                  r          2                            2        d        R              )    −      1    2        (    −    d    +    r    +    R    )    (    d    +    r    −    R    )    (    d    −    r    +    R    )    (    d    +    r    +    R    )  We will now integrate over this d.  K  =      ∫          d      =      R      −      c              d      =      R            (    A    (    R    ,    r    ,    d    )    ⋅                  d        2                    R        2              )  You have to multiply with             d      2              R      2      , since the probability grows with d that the first point picked lies at a specific distance from the center.

In a circle of radius R two points are chosen at random(the points can be anywhere, either within the circle or on the boundary). For a fixed number c, lying between 0 and R, what is the probability that the distance between the two points will not exceed c?

Answered question

Answer & Explanation

New Questions in High school geometry