Question

In the Blade Runner universe, replicants are bioengineered androids that are virtually identical to humans. The “Voight-Kampff” test is designed to distinguish replicants from humans based on their emotional response to test questions. The test designers guarantee an accuracy rate of 90%. In other words, they guarantee that if a replicant is subjected to the test, then the test will correctly label them as a replicant with probability q = 90%. With the remaining probability, the test incorrectly labels the replicant as a human. Similarly, if a human is subjected to the test, then they will be correctly labelled as human with probability q = 90%, and with the remaining probability they will be incorrectly labelled as a replicant. A subject, Leon, is suspected to be a replicant. Your prior probability that Leon is a replicant equals p = 75% and with the remaining probability 1 − p = 25% you suspect Leon is a human. (a) What is the probability that if Leon takes the Voight-Kampff test, the test will label him as a replicant? (b) Leon is subjected to the Voight-Kampff test, and the test labels Leon as a replicant. What is your posterior probability about whether Leon is a replicant or not? (c) Another subject, Deckard, is also suspected to be a replicant, and your prior probability is that Deckard is a replicant with probability p1 = 10% and human with probability 1 − p1 = 90%. Deckard takes the test, and is labelled as a human. What is your posterior probability about Deckard?

Answer 1

(a)

Probability that if Leon takes the Voight-Kampff test, the test will label him as a replicant = P(test replicant)

= P(test replicant | is replicant) P(is replicant) + P(test replicant | is human) P(is human) (Law of total probability)

= 0.90 * 0.75 + 0.10 * 0.25

= 0.7

(b)

Posterior probability about whether Leon is a replicant = P(is replicant | test replicant)

= P(test replicant | is replicant) P(is replicant) / P(test replicant) (Bayes theorem)

= 0.90 * 0.75 / 0.7

= 0.9642857

Posterior probability about whether Leon is a human = P(is human | test replicant)

= P(test replicant | is human) P(is human) / P(test replicant) (Bayes theorem)

= 0.10 * 0.25 / 0.7

= 0.03571429

(c)

Probability that if Deckard takes the Voight-Kampff test, the test will label him as a human = P(test human)

= P(test human | is replicant) P(is replicant) + P(test human | is human) P(is human) (Law of total probability)

= 0.10 * 0.10 + 0.90 * 0.90

= 0.82

Posterior probability about whether Deckard is a replicant = P(is replicant | test human)

= P(test human | is replicant) P(is replicant) / P(test human) (Bayes theorem)

= 0.10 * 0.10 / 0.82

= 0.01219512

Posterior probability about whether Deckard is a human = P(is human | test human)

= P(test human | is human) P(is human) / P(test human) (Bayes theorem)

= 0.90 * 0.90 / 0.82

= 0.9878049