Geometric probability (dealing with integer numbers)Two numbers a and b are chosen from {1,2,…N} with replacement. Let p N be the probability that a 2 + b 2 ≤ N 2 . What is the value of lim N → ∞ p N ?With real numbers, I would just use the geometric approach: π ⋅ N 2 / 4 N 2 so the limit as N → ∞ equals π 4 .

Question

Geometric probability (dealing with integer numbers)Two numbers a and b are chosen from {1,2,…N} with replacement. Let       p    N   be the probability that       a    2    +      b    2    ≤      N    2  . What is the value of       lim          N      →      ∞                  p      N      ?With real numbers, I would just use the geometric approach:             π      ⋅              N        2                    /            4              N      2       so the limit as   N  →  ∞ equals       π    4  .

Cynthia Lester · Accepted Answer

Step 1Let       C    n   denote the number of lattice points on or inside the quarter of the circle       x    2    +      y    2    =      n    2   that lies in the positive quadrant. We can show (by simple counting) that       C    n    =      ∑          k      =      0        n    ⌊            n      2        −          k      2        ⌋  ..Note, here we don&#039;t have to include the points on the x and y axes, so we should subtract a term of O(n) from the above       C    n    .. Furthermore,   x  −  1  &amp;lt;  ⌊  x  ⌋  ≤  x  , hence       D    n    =  #  {  (  x  ,  y  )  :      x    2    +      y    2    ≤      n    2     and   x  ,  y  =  1  ,  2  ,  …  ,  n  } equals      ∑          k      =      1        n              n      2        −          k      2        +  O  (  n  )  .Step 2The required probability is given by       p    n    =            D      n              n      2        =      1    n        ∑          k      =      1        n              n      2        −          k      2        +  O      (          1      n        )    .As   n  →  ∞  , the RHS converges to       ∫    0    1        1    −          x      2        d  x  , which equals   π      /    4  ,, because it calculates the area of a quarter of the unit circle.

Two numbers a and b are chosen from {1,2,…N} with replacement. Let p_N be the probability that a^2+b^2 leq N^2. What is the value of lim_{N rightarrow infty} p_N?

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Answer & Explanation

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