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Math Help - Probability help

  1. #1
    Len
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    Probability help

    Suppose that a point (X_1,X_2,X_3) is chosen at random, that is, in accordance with the uniform probability density function over the following set S:
    S = {(x_1,x_2,x_3) : 0 \leq x_1 \leq 1, for i = 1,2,3}
    <br />
Determine P[(X_1-\frac{1}{2})^2+(X_2-\frac{1}{2})^2+(X_3-\frac{1}{2})^2 \leq \frac{1}{4}

    Not sure how to do this and I have many more similar problems so if someone could give me some help here I may be able to do them all.

    Thanks.
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  2. #2
    Flow Master
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    Quote Originally Posted by Len View Post
    Suppose that a point (X_1,X_2,X_3) is chosen at random, that is, in accordance with the uniform probability density function over the following set S:
    S = {(x_1,x_2,x_3) : 0 \leq x_1 \leq 1, for i = 1,2,3}
    <br />
Determine P[(X_1-\frac{1}{2})^2+(X_2-\frac{1}{2})^2+(X_3-\frac{1}{2})^2 \leq \frac{1}{4}

    Not sure how to do this and I have many more similar problems so if someone could give me some help here I may be able to do them all.

    Thanks.
    0 \leq x_1, 0 \leq x_2, 0 \leq x_3 defines a unit cube.

    \left(X_1-\frac{1}{2}\right)^2 + \left(X_2-\frac{1}{2}\right)^2 + \left(X_3-\frac{1}{2}\right)^2 \leq \frac{1}{4} defines a sphere with centre at \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right) and radius r = \frac{1}{2}. This sphere is inside the cube.

    The probability you want is equal to the proportion of volume of the cube occupied by the sphere, that is, \frac{V_{\text{sphere}}}{V_{\text{cube}}}.
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