Base Rate Fallacy - Mathematical Formalism

Mathematical Formalism

In the above example, where P(T|B) means the probability of T given B, the base rate fallacy is committed by assuming that P(terrorist|bell) equals P(bell|terrorist) and then adding the premise that P(bell|terrorist)=99%. Now, is it true that P(terrorist|bell) equals P(bell|terrorist)?



Well, no. Instead, the correct calculation uses Bayes' theorem to take into account the prior probability of any randomly selected inhabitant in the city being a terrorist and the total probability of the bell ringing:


\begin{align} P(\mathrm{terrorist}|\mathrm{bell}) &= \frac{P(\mathrm{bell} | \mathrm{terrorist}) P(\mathrm{terrorist})} {P(\mathrm{bell})} \\ &= \frac{P(\mathrm{bell} | \mathrm{terrorist}) \times P(\mathrm{terrorist})} { P(\mathrm{bell} | \mathrm{terrorist}) \times P(\mathrm{terrorist}) + P(\mathrm{bell} | \mathrm{nonterrorist}) \times P(\mathrm{nonterrorist})} \\ &= \frac{ 0.99 \cdot (100/1,000,000)} {\frac{0.99 \cdot 100}{1,000,000} + \frac{0.01 \cdot 999,900}{1,000,000}} \\ &= 1/102 \approx 1\% \end{align}

Thus, in the example the probability was overestimated by more than 100 times due to the failure to take into account the fact that there are about 10000 times more nonterrorists than terrorists (a.k.a. failure to take into account the 'prior probability' of being a terrorist).

Read more about this topic:  Base Rate Fallacy

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