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# help with probability

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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

Mar 6, 2020

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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%

Mar 6, 2020