A problem on conditional geometric probabilityThe point (x, y) is chosen randomly in the unit square. What is the conditional probability of x 2 + y 2 ≤ 1 / 4 given that x y ≤ 1 / 16.I started solving this and while calculating I got some very unpleasant numbers but the problem is not marked as difficult in this book where I found it. So I start suspecting that I am having some conceptual mistake.The two curves intersect at x = 2 + 3 / 4 and at x = 2 − 3 / 4Now I have to find a few areas via definite integrals. Is that what this problem is about?

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A problem on conditional geometric probabilityThe point (x, y) is chosen randomly in the unit square. What is the conditional probability of       x    2    +      y    2    ≤  1      /    4 given that   x  y  ≤  1      /    16.I started solving this and while calculating I got some very unpleasant numbers but the problem is not marked as difficult in this book where I found it. So I start suspecting that I am having some conceptual mistake.The two curves intersect at   x  =      2    +          3            /    4 and at   x  =      2    −          3            /    4Now I have to find a few areas via definite integrals. Is that what this problem is about?

micelarnyiz · Accepted Answer

Step 1  x  y  ≤      1    16   iff   2  x  y  ≤      1    8  .Conditional on this,       x    2    +      y    2    ≤      1    4   iff   2  x  y  ≤      1    8  .Conditional on this,       x    2    +      y    2    ≤      1    4   iff       x    2    +  2  x  y  +      y    2    =  (  x  +  y      )    2    ≤      3    8   iff   x  +  y  ≤            3      8      .Step 2Now do you know how to calculate the probability that the sum of two random numbers is less than a given number?

nascarchic839e9 · Accepted Answer

Explanation:Suppose that   x  y  ≤      1    16    .. Then       x    2    +      y    2    ≤      1    4      ⟺    (  x  +  y      )    2    ≤      3    8  . This should lead to an easier way to approach it.

The point (x, y) is chosen randomly in the unit square. What is the conditional probability of x^2+y^2 leq 1/4 given that xy leq 1/16

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