Question

The quality control manager at a bakery is inspecting a batch of 100 chocolate chip cookies. If the production process is in control, the mean number of chocolate chips per cookie is 6.0. Assuming the process is Poisson distributed, how many cookies should the manager expect to discard from the batch if the policy is that each cook must have at least four chocolate chips?

Answer 1

Answer:

0.84882

Step-by-step explanation:

Given:-

The production process is in control is number of chocolate chips in a cookie. The mean number of chocolate chips per cookie is 6.0.

                                           λ = 6.0

- We define a random variable (X) that denotes the number of chocolate chips inspected on a cookie follows a poisson distribution:

                                          X ~ Po ( 6.0 )

Find:-

 how many cookies should the manager expect to discard from the batch if the policy is that each cook must have at least four chocolate chips?

Solution:-

- We will use the probability mass function of Poisson variate (X). Given:

                              $P ( X = x) = \frac{lambda^k*e^-^l^a^m^b^d^a}{k!} \\\\P ( X = x) = \frac{6^k*e^-^6}{k!}$

Where,                   k = 0 , 1 , 2 , 3 , 4 , 5 , 6 , .... , n = 100

- The required probability is P ( X ≥ 4 ); Using the pmf function we have:

                 $P ( X \geq 4 ) = 1 - P ( X < 4 )\\\\P ( X \geq 4 ) = 1 - [ \frac{6^0*e^-^6}{0!} + \frac{6^1*e^-^6}{1!} + \frac{6^2*e^-^6}{2!} + \frac{6^3*e^-^6}{3!} ]\\\\P ( X \geq 4 ) = 1 - [ 0.00247 + 0.01487 + 0.04461 + 0.08923 ]\\\\P ( X \geq 4 ) = 1 - [ 0.15118 ] = 0.84882$