You write a computer vision application that identifies peop

You write a computer vision application that identifies peoples’ faces with an accuracy of 90%. Your software is deployed in a security system that screens 50 people per day. For each of the questions below, please give ALL of the following information: (1) the formula to compute the answer, (2) the R command that computes the answer, and (3) the final number for the probability.

(a) What is the probability that your system will have exactly 8 errors in a day?

(b) What is the probability that your system will have 5 or fewer errors in a day?

(c) What is the probability that your system will have a “perfect day” (no errors)?

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 8 ) = ( 50 8 ) * ( 0.1^8) * ( 1 - 0.1 )^42
= 0.06428
b)
P( X < = 5) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 50 5 ) * 0.1^5 * ( 1- 0.1 ) ^45 + ( 50 4 ) * 0.1^4 * ( 1- 0.1 ) ^46 + ( 50 3 ) * 0.1^3 * ( 1- 0.1 ) ^47 + ( 50 2 ) * 0.1^2 * ( 1- 0.1 ) ^48 + ( 50 1 ) * 0.1^1 * ( 1- 0.1 ) ^49 + ( 50 0 ) * 0.1^0 * ( 1- 0.1 ) ^50
= 0.61612
c)
P( X = 0 ) = ( 50 0 ) * ( 0.1^0) * ( 1 - 0.1 )^50
= 0.00515