An archer shoots an arrow at a circular target of radius 50 cm. The arrow pierces the target at a random position (X,Y), measured in centimeters from the center of the disk at position (X,Y) = (0,0). The bullseye is a solid black circle of radius 2 cm, at the center of the target. Calculate the probability P[B] of the event that the archer hits the bullseye under each of the following models: a) X and Y are iid continuous uniform (-50,50) random variables. b) The PDF fX,Y(x,y) is uniform over the 50 cm circular target. c) X and Y are iid Gaussian (μ=0, σ=10) random variables.

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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An archer shoots an arrow at a circular target of radius 50 cm. The arrow pierces the target at a random position (X,Y), measured in centimeters from the center of the disk at position (X,Y) = (0,0). The bullseye is a solid black circle of radius 2 cm, at the center of the target. Calculate the probability P[B] of the event that the archer hits the bullseye under each of the following models:

a) X and Y are iid continuous uniform (-50,50) random variables.

b) The PDF fX,Y(x,y) is uniform over the 50 cm circular target.

c) X and Y are iid Gaussian (μ=0, σ=10) random variables.

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