Can You Find The Probability?
I am not sure if there’s any real world application to this. That said, I do know that these puzzles are a great way to kill time and exercise our brains from time to time.
Perhaps I am indirectly lowering the strain on the public health system by getting all of us to stay sharp!
Solution
When you pick three random points on a circle and form a triangle, the triangle contains the center of the circle if and only if the three points are not all contained within any semicircle.
Here are 4 examples where all three points lie within some semicircle. We can see that the triangle will sits on one side of the circle and cannot enclose the center.
So let’s turn our attention to finding the probability that three random points on a circle all lie in some semicircle.
Let’s say we fix one point on the circle.
The first point is completely arbitrary, it doesn’t matter where we pick it.
Now the other two points must lie within the semi-circle that contains the first point.
A semi-circle is half the circle, this means each point independently has probability 1/2 of landing in that semicircle.
In this example, our semicircle is the upper half of the circle. Try and imagine those two blue points being put in any of the positions of the circumference of the semicircle.
You will find that all triangles formed from these points do not contain the center.
So for each individual starting point, the probability that the triangle don’t contain the center is
But wait! Because any of the 3 points can be the starting point, the probability that all those points are in some semi-circle is actually 3 times that.
This means the probability that the triangle formed from 3 arbitrary points does contain the center must be
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