I am sorry that I am not quite familiar with Latex, so all the mathematical formula texted here is ugly.
A formula related to conditional probability is sometimes overlooked however sometimes it's quite convenient to use it. The equation is
P(A, B | C) = P(A | B, C) * P(B | C) (1)
The proof is direct, since
P(A, B | C) = P(A, B, C) / P(C) (2)
and
P(A | B, C) = P(A, B, C) / P(B, C) (3)
P(B | C) = P(B, C) / P(C) (4)
Multiply the right sides of both (3) and (4) will lead to the right side of (2), so the formula holds
A formula related to conditional probability is sometimes overlooked however sometimes it's quite convenient to use it. The equation is
P(A, B | C) = P(A | B, C) * P(B | C) (1)
The proof is direct, since
P(A, B | C) = P(A, B, C) / P(C) (2)
and
P(A | B, C) = P(A, B, C) / P(B, C) (3)
P(B | C) = P(B, C) / P(C) (4)
Multiply the right sides of both (3) and (4) will lead to the right side of (2), so the formula holds
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