The quality control manager of Marilyn\'s Cookies is inspecting a batch of choco

Question

The quality control manager of Marilyn's Cookies is inspecting a batch of chocolate-chip cookies that has just been baked. If the production process is in control, the mean number of chip parts per cookie is 6.5 .Complete parts (a) through (d).

a. What is the probability that in any particular cookie being inspected fewer than five chip parts will be found?

b. What is the probability that in any particular cookie being inspected exactly five chip parts will be found?

c. What is the probability that in any particular cookie being inspected five or more chip parts will be found?

d. What is the probability that in any particular cookie being inspected either four or five chip parts will be found?

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a.
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-6.5 * 5 ^ 4 / 4! + e^-6.5 * ^ 3 / 3! + e^-6.5 * ^ 2 / 2! + e^-6.5 * ^ 1 / 1! + e^-6.5 * ^ 0 / 0!
= 0.2237

b.
P( X = 5 ) = e ^-6.5 * 6.5^5 / 5! = 0.1454

c.
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-6.5 * 5 ^ 4 / 4! + e^-6.5 * ^ 3 / 3! + e^-6.5 * ^ 2 / 2! + e^-6.5 * ^ 1 / 1! + e^-6.5 * ^ 0 / 0!
= 0.2237
P( X > = 5 ) = 1 - P (X < 5) = 0.7763

d) P(x=4) or P(X=5) = 0.1454+0.1118 = 0.2572

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