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Math Help - probability of most like cause?

  1. #1
    Yan
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    probability of most like cause?

    an explosion at a construction site could have occurred as the result of static electricity, malfunctioning of equipment, carelessness, or sabotage. Interviews with construction engineers analyzing the risks involved led to the estimates that such an explosion would occur with probability 0.25 as a result of static electricity, 0.20 as a result of malfunctioning of equipment, 0.40 as a result of carelessness, and 0.75 as a result of sabotage. It is also felt that the prior probabilities of the four causes of the explosion are 0.20, 0.40, 0.25, and 0.15. Based on all this information, what is

    (a) the most likely cause of the explosion;
    (b) the least likely cause of the explosion?
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    Quote Originally Posted by Yan View Post
    an explosion at a construction site could have occurred as the result of static electricity, malfunctioning of equipment, carelessness, or sabotage. Interviews with construction engineers analyzing the risks involved led to the estimates that such an explosion would occur with probability 0.25 as a result of static electricity, 0.20 as a result of malfunctioning of equipment, 0.40 as a result of carelessness, and 0.75 as a result of sabotage. It is also felt that the prior probabilities of the four causes of the explosion are 0.20, 0.40, 0.25, and 0.15. Based on all this information, what is

    (a) the most likely cause of the explosion;
    (b) the least likely cause of the explosion?
    Bayes' theorem tells us that:

    p(cause|explosion)p(explosion)=p(explosion|cause)p (cause)

    So we are left with:

    p(cause|explosion)=p(explosion|cause)p(cause)/p(explosion)

    Now we don't know p(explosion) but you don't need to know it to answer the questions as it is a common factor, that is we have:

    p(cause|explosion)=k p(explosion|cause)p(cause)

    for some k and to answer the questions you just need to evaluate p(explosion|cause)p(cause) for each cause.

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