We have a stick of length L. The stick got broken at (n-1) randomly chosen points (lengths of parts can be non-integer or floating point numbers also) so we get n parts. We need to find the probability that these n pieces can form a n sided polygon.
Input : L = 5 n = 3 Output : 0.25 We need to cut rope of length 5 into three parts.
First we need to find the condition when n lengths can form a n sided polygon. Let us consider a triangle, we know for a triangle the length of the largest side must be smaller than the sum of the lengths of other sides. Similarly for a n sided polygon the the length of the largest side must be less than the sum of the other (n-1) sides. Why? Suppose we break the stick into two equal halves. We further break one of the halves into (n-1) parts. We can never place them such that they make a closed polygon. (Actually the best we can do is to make 2 parallel lines). So we just need to find the probability that no part has the length greater than or equal to L/2.
Now we need to work on the probability. There are many ways to calculate the required probability we will use a geometric approach. Consider a circle of perimeter L. We place n points on the perimeter. The probability that they lie on the same semicircle is . Please refer this link for more information, let us denote it by P(E).
This probability is actually same as breaking the stick such that at least one part is of length L/2. But we want just the complement of this event hence our answer is
- Probability of cutting a rope into three pieces such that the sides form a triangle
- Number of ways to cut a stick of length N into K pieces
- Apothem of a n-sided regular polygon
- Length of Diagonal of a n-sided regular polygon
- Area of a n-sided regular polygon with given Radius
- Determine the position of the third person on regular N sided polygon
- Side of a regular n-sided polygon circumscribed in a circle
- Find number of diagonals in n sided convex polygon
- Area of a n-sided regular polygon with given side length
- Area of largest Circle inscribe in N-sided Regular polygon
- Number of triangles formed by joining vertices of n-sided polygon with one side common
- Number of triangles formed by joining vertices of n-sided polygon with two common sides and no common sides
- Count pieces of circle after N cuts
- Maximum number of pieces in N cuts
- Minimum area of a Polygon with three points given
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