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Spam and Bayes' theorem

I divide my email into three categories: A1 = spam. A2 = low priority, A3 = high priority.

I find that: P(A1) = .7
P(A2) = .2
P(A3) = .1
Let B be the event that an email contains the word "free".
P(B|A1) = .9
P(B|A2) = .01
P(B|A3) = .01
I receive an email with the word "free". What is the probability that it is spam?
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Comments

  1. Dieu-Thu LeFebruary 9, 2012 at 7:34 PM

    1. The law of total probability:
    P(B) = sum_{j=1}^{k} P(B|Aj) * P(Aj)
    with A1 -> Ak: a partition of the sample space omega

    2. Bayes' Theorem:
    P(Ai|B) = P(B|Ai) * A{Ai) / sum_{j} P(B|Aj) * P(Aj)

    Hence:
    P(A1|B) = P(A1B)/P(B) = P(B|A1) * P(A1) / sum_{j} P(B|Aj) * P(Aj)
    = 0.995

    ReplyDelete

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