Suppose you break a stick into three pieces and try to assemble them into a triangle. If you choose the two breaking points at random points in the interval, the theoretical probability that the pieces are able to form a triangle is 14 or 25%. To see how well this theoretical probability works in practice, select two points on the line below at random by clicking on the image then click 'Assemble Triangle' to see if a triangle can be made out of those pieces. To try to make another triangle yourself, click 'Start a new triangle' and select new breaking points. You can also let Maple generate random sets of breaking points.
Explanation of probability
Assume the endpoints of the stick are at the points 0 and 1.
Let x and y be the two breaking points, chosen at random from the values in the interval 0, 1.
Let us assume at first that y> x. The three sides of the triangle have lengths x, y−x, and 1−y. The three conditions that must be satisfied for a triangle to be formed are thus:
x+y−x > 1−y,
y−x+1−y > x,
1−y + x> y−x.
Simplifying these inequalities gives us:
x < 12, y > 12, and y−x < 12.
We can rearrange this to give:
x < 12, 12 < y < 12+x.
The total probability for this case can thus be given by the integral:
P=∫012∫1212+x ⅆyⅆx=∫012x ⅆx = 18.
Considering that the other case (that y < x) is equally likely, the total probability that a triangle can be formed is 18+18= 14.
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