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Can We Use Bayes Theorem in Predicting Bomb Threats?

http://www.psychologyinaction.org/2012/10/22/bayes-rule-and-bomb-threats/

Most of us are familiar with Bayes Theorem.  It’s a theorem that states that the probability of any event is based on the likelihood of conditions relating to the original event.  Mathematically stated Bayes Theorem appears as the following equation:

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Where P(A) and P(B) are the probabilities of two separate but related events.  P(B|A) is the probability of event B given the likelihood of event A.  And P(A|B) is the probability of event A given the likelihood of event B.

In terms of the use of Bayes Theorem for airport security, we are all familiar with the x-ray machines used in airports to detect bombs and other substances. Let’s assume approximately .000001% of the people in airports are actually carrying bombs and these x-ray machines have a 95% chance of identifying a bomb in a passenger’s bag when one actually exists and a 5% chance of identifying a bomb in a passenger’s bag when one does not exist.

Using Bayes Theorem and the numbers given above, let’s try to calculate the probability the x-ray machine correctly identifies a bomb.

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This is an astounding number; .000019 means that approximately .002% of the time does the x-ray machines correctly predict a bomb when one exists.  Psychologically one would think that the accuracy of the machine allows for this percentage to be much higher, however, mathematically viewing these number could explain the reason that we are so heavily checked at security.

The article listed above explains that the prominence of an event such as an explosion caused by a bomb affects how we think about likelihoods.  Bayes Theorem allows us to mathematically rationalize these likelihoods and our fears by using correct posterior probabilities instead of incorrect ones created by our fear.

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