Suppose we have a face recognition system with an accuracy of 90%. Suppose it is deployed in a security system that observes 40 people. Compute the following probabilities
(a) It fails for exactly 6 people.
(b) It fails for at most 2 people.
(c) It fails for at least 8 people
These are binomial probabilities
(a) It fails for exactly 6 people =
C(40, 6) (.10)^6 (.90)^34 ≈ 10.68%
(b) It fails for at most 2 people =
P(It fails for 0 people) + P(It fails for 1 person) + P (It fails for 2 people) =
(.90)^40 + C(40,1) (.10)(.90)^39 + C(40,2)(.10)^2(.90)^38 ≈ 22.29%
(d) It fails for 8 people at most =
1 - [ P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) ] =
1 - [(.90)^40 + C(40,1) (.10)(.90)^39 + C(40,2)(.10)^2(.90)^38 + C(40,3) (.10)^3(.90)^37 + C(40,4)(.10)^4(.90)^36 + C(40,5) (.10)^5(.90)^35 + C(40,6)(.10)^6(.90)^34 + C(40,7)(.10)^7(.90)^33 ] ≈
1 - [ .2229 + .7353 ] ≈ 4.18%