Probability circle to be in a circleWe have circle with radius R and inside it we choose at random 2 points A and B.Find the probability the circle with center A and radius AB to be inside the circle(It is possible both circles to be tangent)What I've concluded : the first point A has range from [0;R] and the second point B must have range from [ 0 ; R − O A ] .But how should I find the probability?

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Probability circle to be in a circleWe have circle with radius R and inside it we choose at random 2 points A and B.Find the probability the circle with center A and radius AB to be inside the circle(It is possible both circles to be tangent)What I&#039;ve concluded : the first point A has range from [0;R] and the second point B must have range from   [  0  ;  R  −  O  A  ] .But how should I find the probability?

Prezrenjes0n · Accepted Answer

Step 1I can propose a little different approach. We want to choose 2 points (call them X,Y) inside a circle of radius R. Due to circle being rotation-invariant, we can always rotate so that X lies on   [  0  ,  R  ]  ×  {  0  }  ⊂            R        2  . So that we need to find the distribution of   Z  :=      |        |    X      |        |  . We have       F    Z    (  t  )  =            π              t        2                    π              R        2                  χ          [      0      ,      R      ]        (  t  )  +      χ          (      R      ,      +      ∞      )        (  t  ), so that the density is       g    Z    (  t  )  =      2          R      2        t      χ          [      0      ,      R      ]        (  t  ).Let   A  ∈      F   be an event - circle with center X and radius       |        |    Y  −  X      |        |   is inside our circle of radius R.Step 2Then we have       P    (  A      |    Z  )  =            π      (      R      −      Z              )        2                    π              R        2              =            (      R      −      Z              )        2                    R      2      And by total expectation      P    (  A  )  =      E    [      P    (  A      |    Z  )  ]  =      1          R      2            E    [  (  R  −  Z      )    2    ]  =      1          R      2            ∫    0    R    (  R  −  z      )    2    z      2          R      2        d  z  =      2          R      4            ∫    0    R        s    2    (  R  −  s  )  d  s  =      2          R      4        (            R      4        3    −            R      4        4    )  =      1    12

We have circle with radius R and inside it we choose at random 2 points A and B.Find the probability the circle with center A and radius AB to be inside the circle(It is possible both circles to be tangent)

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