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When the motion of a microscopic particle in a liquid or a gas is observed, it
is seen that the motion is irregular because the particle collides frequently
with other particles. The probability model for this motion, which is called
Brownian motion, is as follows: A coordinate system is chosen in the liquid or
gas. Suppose that the particle is at the origin of this coordinate system at
time t = 0; and let (X,Y,Z) denote the coordinates of the particle at any time
t > 0. The random variables X,Y,and Z are i.i.d. and each of them has a normal
distribution with mean 0 and variance (sigma^2)t. Find the probability
that at time t = 2 the particle will lie within a sphere of radius 3.5sigma
whose center is at the origin. The closest value is:
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