Research task (STEM):
I want a comprehensive research paper on the history of probability problems involving circles and stochastically selecting points or lines across the circle's circumference. As an introduction, start off by explaining in depth the solution to the following problem: "7 points are randomly chosen on the circumference of a circle, what is the probability that they all lie within a semi-circular arc (that is an arc half the circle's size)?" and then provide related problems and their solutions and a general solution methodology for this style of problem. Then provide a history of interesting problems along these specifications, specifically I would like you to give me an overview of the lazy caterer's sequence and the n cuts to a cake problem. 

Evaluation rubric:
criterion: The response states the exact probability that 7 random points all lie within some semicircle as 7/64 (≈0.109375).
weight: 5
axis: Explicit Criteria

criterion: The response derives the solution (p = 7/64) result by fixing one point (or an equivalent symmetry argument) and notes the general formula P(n)=n/2^(n−1) for n points.
weight: 5
axis: Implicit Criteria

criterion: The response provides at least two additional probability problems involving random points, lines, or chords on a circle and supplies their complete solutions (e.g., when is a random triangle on a circle acute, the probability all n points lie on arc of fixed length, probability a randomly inscribed k-gon contains the center, probability the center lies in the convex hull of n points).
weight: 5
axis: Explicit Criteria

criterion: The response explains a general solution methodology for this class of circle-probability problems (e.g., rotational symmetry, fixing a reference point, combinatorial counting, order statistics on the circle).
weight: 5
axis: Explicit Criteria

criterion: The response provides a historical overview that names and briefly describes at least three notable circle-based probability problems (e.g., Bertrand’s Paradox, Buffon’s Needle on a circle, the random chord problem, Sylvester's four-point problem for circles).
weight: 5
axis: Explicit Criteria

criterion: The response defines the lazy caterer’s sequence, provides its closed-form formula L_n = n(n+1)/2 + 1, and lists at least the first five terms (1,2,4,7,11).
weight: 5
axis: Implicit Criteria

criterion: The response describes the “n straight cuts to a cake” problem and states that the maximum number of pieces produced by n cuts follows the lazy caterer’s sequence.
weight: 5
axis: Synthesis of Information

criterion: The response is presented in the form of a research paper with clearly labeled sections, including an Introduction that starts with the 7-point semicircle problem, plus separate sections for Related Problems, Methodology, Historical Background, Lazy Caterer’s Sequence / Cake-Cut Problem, and a Conclusion (≥5 section headings in total).
weight: 5
axis: Instruction Following

criterion: The response omits unrelated mathematical topics (e.g., probabilities on polygons as opposed to circles, Pólya's enumeration theorem, permutation groups, topology) and stays within the scope requested by the user.
weight: 4
axis: Implicit Criteria

criterion: The response must use mathematical notations to express probability calculations and geometric relationships, including formal definitions and equations for the circle problems discussed.
weight: 5
axis: Implicit Criteria

criterion: The response provides at least two frameworks to solve the 7-point problem (e.g., dividing the circle into 2𝑘 equal sectors, broken stick approach, anchor point, brute force solution treating all points as random variables).
weight: 3
axis: Implicit Criteria

criterion: The response details at least two common pitfalls or incorrect deductions when solving the 7-point problem (e.g., failing to consider any of the n points as "leading", failing to treat the placement of points as independent, treating the circumference of the circle as an important variable, proceeding by induction but not establishing a base case).
weight: 4
axis: Synthesis of Information

criterion: The response explains briefly how f(n) = f(n-1) + n reduces down to (n^2+n+2)/2 for the caterer's sequence.
weight: 4
axis: Implicit Criteria

criterion: The response mentions OEIS, the Online Encyclopedia of Integer Sequences.
weight: 2
axis: Instruction Following

criterion: The response connects the caterer's sequence to Pascal's triangle (i.e., the sequence is the sum of the first three entries in row n of Pascal’s triangle).
weight: 4
axis: Synthesis of Information

criterion: The response gives the cake number's formula as (n^3+5n+6)/6.
weight: 5
axis: Implicit Criteria

criterion: The response gives the recurrence relation Cake(n) = Cake(n-1) + caterer(n-1).
weight: 5
axis: Implicit Criteria

criterion: The response generalizes the caterer's sequence to N dimensions correctly as R(n,d) = sum over i from 0 to d of n choose i.
weight: 4
axis: Implicit Criteria

criterion: The response highlights the importance of exclusivity in the 7 point problem solution (i.e., we count configurations by anchoring each semicircle at a sampled point).
weight: 4
axis: Synthesis of Information

criterion: The response mentions at least two related integral statistics ideas (independence, Principle of Inclusion Exclusion, Bayes theorem, Markov chains) in context of the problems discussed.
weight: 3
axis: Implicit Criteria

criterion: The response presents the historical problems chronologically.
weight: 4
axis: Instruction Following

criterion: The response mentions the problem of Buffon's Needle (i.e., we drop a needle of length L onto a floor marked with parallel lines spaced D apart (D ≥ L). What’s the probability p the needle crosses a line?). 
weight: 3
axis: Implicit Criteria

criterion: The response presents the formulae for the Lazy Caterer Sequence and the Cake Numbers without any derivation or explanation of their terms.
weight: -5
axis: Implicit Criteria

criterion: The response provides the solution to Buffon's Needle problem (i.e., p = 2L/(\pi D)).
weight: 3
axis: Implicit Criteria