Failure to Take Base Rates Into Account

One way to understand probability asserts that the probability of an event, P(A), equals the number of possible outcomes (or ways the world could be) that make A true, f, divided by the total number of possible outcomes, n.  Thus,  P(A) = f/n. To update our probability in light of new evidence, we can use Bayes’ Theorem (this formulation is limited to two mutually exclusive and jointly exhaustive events).  In the formulation below, A₁ is the first possible event, A₂ is the second possible event.   B is our new evidence, P(B given A₁) is the probability of our new evidence given that A₁ is true, and P(B given A₂) is the probability of our new evidence given that A₂ is true.      

Thus, we can think of Bayes' theorem as recalculating f/n: In the original probability for A₁ would equate A₁ with f and equate A₁ + A₂ with n.  That is, the probability of P(A₁)= P(A₁)/P(A₁) + P(A₂), i.e., P(A₁)/P(A₁) or P(A₂).  So, to modify the probability of A₁ given B we calculate the probability of A₁ and the probability of B given A₁, or P(A₁) x P(B given A₁) divided by the probability of A₁ and B given A₁ plus the probability A₂ and B given A₂, or [P(A₁) x P(B given A₁)] + [P(A₂) x P(B given A₂)].

In the case from the slides, the probability of being a blue gang member, A₁, simpliciter is .15 (15%).  The probability of being a green gang member, A₂, simpliciter is .85 (85%).  The probability of eye witness testimony, B, given that the shooter is a blue,  P(B given A₁), is .8 (80%).   The probability of eye witness testimony, B, given that the shooter is a green,  P(B given A₂), is .2 (20%).  Thus, the probability of the shooter being a blue gang member in light of the eyewitness testimony, i.e., P(A₁ given B), is given by plugging the numbers into Bayes' theorem: