D. How isBayes theorem important in avoiding fallacies commonly made in law andcourtrooms? Base rate fallacy If presented with related generic, generalinformation and specific information (information concerning a particular case),the mind tends to ignore the former and focus on the latter.1This can be avoided using Bayes’ Theorem, as it takes into account priorprobability (whichis the probability assessed before making reference to certain relevantobservations, especially subjectively or on the assumption that all possibleoutcomes be given the same probability) and posterior probability (which is the conditional probability that is assigned after the relevant evidence orbackground is taken into account) Example:Drunk drivers measured by a breathalyzer. Theconditions are as follows: ?      Breathalyzersdisplay a false result in 5% of the cases tested.?      They never failto detect a person who is truly drunk.

?       drivers on the road are drunk.  If a policeman stops a driver at random and the breathalyzer displays that the driver is drunk, what is the probability that the driver is truly drunk??     For the 1 driver who istruly drunk, there is a 100% chance of a truepositive test result as the second condition stated. Thus, the true positive test result is 1. ?     For the other 999 driverswho are not drunk, there is a 5% chance of falsepositive test results, as the first condition stated. Thus, the false positive test results is 49.95(999 x 5%)From this, we can calculatethe probability that one of the drivers among the 50.95 (1 + 49.

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95) positivetest results, who is really drunk:  = 0.019627 This can be confirmed withBayes Theorem. Notations:D = drunk S = sober B = breathalyzer indicates that thedriver is drunk Result Drunk Sober Test Positive 1 0.

05 Test Negative 0 0.95 Historic Data 0.001 0.999  Notation form:P(D) = 0.001P(S) = 0.999P(B|D) = 1.00P(B|S) = 0.

005Figure 1: Venn Diagram depictingprobability of driver being drunk or sober and results from breathalyzer.  The probability of thedriver being drunk, given that the breathalyzer indicates that h/she is drunkis represented by p(D|B).According to Bayes’ theorem:P (D|B) = From that we can arrive atp(D):P(D) = P(B|D)P(D) + P(B|S) P(S)  Computing above values toobtain value of P(D) (refer to table and following notationform):         P(D) = (1.00 x 0.001) + (0.05 x0.999)                      = 0.

05095 Computing all values intoBayes’ Theorem formula:P(D|B) =               = 0.0196271 Woodcock, S. (April 5, 2017) Paradoxes of probability and other statistical strangeness

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