Given uniformly randomly chosen x on the interval (1,5) and y on the interval (1,5) find the probability of each:

[x] + [y] = [x+y]

[x] - [y] = [x-y]

[x] * [y] = [x*y]

[x] / [y] = [x/y]

Where [x] is the floor function, the greatest integer less than or equal to x.

(In reply to solutions for first three by Charlie)

There are only 8 unit squares where this is possible at all: those where [x] = [y], [x] = 2[y], [x] = 3[y] and [x] = 4[y]

The overall area of the 16 unit squares is of course 16.

There are 4 squares where [x] = [y], each of which is half good, where x>y, as there is nowhere in any such square where x>2y. So the total good area within such squares is 4*(1/2) = 2.

There are 2 squares where [x] = 2[y]. In each such area, the good area is 1/4 unit, so the total area added is 1/2.

There is 1 square where [x] = 3[y], and in it, 1/6 unit is good area, so this adds 1/6 to the total good area.

There is 1 square where [x] = 4[y], and in it, 1/8 unit is good area, adding 1/8 to the total of good area.

So the probability is (2 + 1/2 + 1/6 + 1/8) / 16 = 67/384 ~= 0.1744791666666666666.

Posted by Charlie on 2007-05-18 11:56:35