Question

(Legal reasoning): Suppose a crime has been committed. Blood is found at the scene for which there is no innocent explanation. It is of a type which is present in 1% of the population.
a. The prosecutor claims: “There is a 1% chance that the defendant would have the crime blood type if he were innocent. Thus there is a 99% chance that he is guilty”. This is known as the prosecutor’s fallacy. What is wrong with this argument?
b. The defender claims: “The crime occurred in a city of 800,000 people. The blood type would be found in approximately 8000 people. The evidence has provided a probability of just 1 in 8000 that the defendant is guilty, and thus has no relevance.” This is known as the defender’s fallacy. What is wrong with this argument?

Answer 1

The prosecutor argues that there is only a 1% chance that the accused would have blood of thistype (event A, say) if innocent (event B, say) and concludes that the accused is guilty. The prosecutor isreasoning in terms of P(A—B) (1%) and P(not A—B) (99%) when it is P(B—A) that is relevant.

For a randomly chosen person, let T be the event ‘person chosen has blood of this type’ and let G bethe event ‘person chosen is guilty’. The figure quoted by the defender isP(G|T), whereas the probabilityof interest isP(accusedisguilty). The argument of the defender would make sense if the accused wasrandomly chosen from the crowd of the 8000 people having this blood type.The accused is not a randomly chosen person, and has additional connections to the case, beyondthe blood type. The blood type is an additional piece of evidence, that needs to be taken into account inconjunction with the other pieces of information