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Understanding Probabilities through a Dart Experiment: A Model for Hit Distance on a Disc

This document presents a probability model based on the dart experiment, where a dart hits a disc of radius ( r ) at a random point. It explores the distance ( X ) from the hitting point to the center of the disc. The cumulative distribution function ( F(b) ) is analyzed, revealing that ( P(0 < X leq r/2) = 1/4 ) and ( P(r/2 < X leq r) = 3/4 ), regardless of the disc's radius. This example provides a practical application of probability in understanding random distributions within geometric constraints.

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Understanding Probabilities through a Dart Experiment: A Model for Hit Distance on a Disc

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  1. CIS 2033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007Slides by Longin Jan Latecki C5:

  2. 5.1 Darts Example Suppose we want to make a probability model for an experiment: A dart hits a disc of radius r in a completely arbitrary way. We are interested in the distance X from the hitting point to the center of the disc. Since distances cannot be negative, we have F(b) = P(X ≤ b) = 0 when b < 0. Since the object hits the disc, we have F(b) = 1 when b > r. b r

  3. Compute for the darts example the probability thatn0 < X ≤ r/2, and the probability that r/2< X ≤ r. We have P(0 < X ≤ r/2) = F(r/2) − F(0) = (1/2)2 − 02 = 1/4, and P(r/2 < X ≤ r) = F(r)−F(r/2) = 1−1/4 = 3/4, no matter what the radius of the disc is.

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