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angel.white · #23 November 14th, 2007, 09:59 PM

Okay, I think I have found a way to calculate it, but I would need computer software to make it work

so we need to know the probability that $B^{2}-4A>0$ is true. Lets modify the equation to make it more workable.

$B^{2}-4A>0$

Add 4A
$B^{2}>4A$

Square root (B must be positive because it is taken from a set of positive integers)
$B>2\sqrt{A}$

So now we need to find the probability that this is true for A and B are elements of {1,2,3,...,2007}

So if we assume, A we can figure out the probability for this to be true for B, based on some specific A. In this case, $2007-floor(2\sqrt{A})$ will give us the number of possibilities that B can be which will satisfy the equations. Then divide that by 2007 (the total number of possibilities that B can be, including those that do and those that do not satisfy the equations), and that will give you the percentage that the equations can be satisfied. Meaning cd=A and c+d=B is true (if A is true).

So for any A, the probability that it has real factors whose sum is B is
$\frac{2007-floor(2\sqrt{A})}{2007}$

And so because any given A has a 1 in 2007 chance of being true, then we can multiply that probability against the probability that B will be true for that A, and then count the total probabilities for each A.
$\frac{1}{2007}\cdot\frac{2007-floor(2\sqrt{A})}{2007}$

$\frac{2007-floor(2\sqrt{A})}{4028049}$

Now we just need to count them all. I'm not very familiar with summation notation, but I think this is correct

$\sum^{2007}_{i=1}\frac{2007-floor(2\sqrt{i})}{4028049}$

As i cycles through every value, it will calculate the odds for an A of that value having a B that will satisfy cd=A and c+d=B, then add it to the running total, and when it is done there should be a percentage, between .96 and 1

Okay, so I don't actually know how to do that, (well, I know what it means, but I don't care to sit here for 2007 iterations and calculate the odds) but I think that it is the answer.

Does anyone know an online site which can do this type of equation? If not, perhaps someone can create a quick program to do it (I used to know how to, but it's been too long

)

Edit: Looks like if you have a graphing calculator, you can do it Summation (Sigma) Notation Using the Graphing Calculator I, unfortunately, do not have a graphing calculator, but if any of you would plug it in for me, I'd +rep you ^_^ (that's worth like an entire point)