• Minsol Kim
• Bhargav Solanki

This repository hosts our group's submission for Task Sheet 4 of Collective robotics and scalability.

file name: buffon.py

Buffon's Needle problem is a classic probability puzzle that involves dropping a needle of length b onto a plane ruled with parallel lines spaced s apart. The goal is to determine the probability P that the needle will intersect one of the lines and use this probability to approximate π.

• Simulation: The simulation involves dropping a needle on a single line segment of width s and checking if it intersects based on the needle’s angle and distance from the center to the nearest line.
• Intersection Probability: The intersection probability P is calculated and used to estimate π according to the formula: [ \pi \approx \frac{2b}{sP} ]

• simulate_buffon_needle(n, b, s): Simulates n needle drops, returning the intersection probability.
• estimate_pi(P, b, s): Estimates π using the intersection probability.
• experiment_and_plot(): Conducts experiments and plots the standard deviation.
• confidence_intervals(): Computes and plots the confidence intervals.
• outside_confidence_ratio(): Measures and plots the ratio of experiments outside the confidence interval.

Problem: In Task 5.1.d, the ratio of experiments for which the true probability was outside the 95% confidence interval was always 0. This suggests that the estimated probabilities always fell within the expected range, which is statistically unlikely. Also, we reduced the number of experiments from 10000 to 100 since running the code was too slow.

1. Given the random nature of the experiments, it's possible that the simulation results are showing a rare scenario where the estimates are unusually accurate. However, this should not consistently result in a ratio of 0 over many trials.

2. There may be a bug or incorrect implementation in calculating the 95% confidence intervals. The intervals may be too wide, consistently covering the true probability.

file name: local_sampling.py

plot results: task5.2_plot.png

This task simulates a swarm of robots uniformly distributed over a unit square. Each robot is randomly assigned a color, either black or white, with equal probability. The goal is for each robot to estimate the overall number of black robots in the swarm based on local observations of its neighbourhood, defined by a sensor range r= [0.1, 0.2, 0.3, 0.4, 0.5].

Consequences: The robot swarm's effectiveness may depend on an accurate estimate of the number of black robots, so sensor range and swarm are crucial. The larger the sensor range better the accurate and reliable estimates.