Answer to Question #125773 in Statistics and Probability for Ayeshah Babar

Question

According to Chemical Engineering Progress
(November 1990), approximately 30% of all pipework
failures in chemical plants are caused by operator error.
(a) What is the probability that out of the next 20
pipework failures at least 10 are due to operator
error?
(b) What is the probability that no more than 4 out of
20 such failures are due to operator error?
(c) Suppose, for a particular plant, that out of the random sample of 20 such failures, exactly 5 are due
to operator error. Do you feel that the 30% figure
stated above applies to this plant? Comment.

Answer 1

X: number of failures due to operator error. P=0.3 , n=20 . 𝑿~𝑩𝒊𝒏𝒐𝒎𝒊𝒂𝒍 (𝟐𝟎, 𝟎. 𝟑) 

𝒇(𝒙) = C_{𝟐𝟎,𝒙} (𝟎. 𝟑) ^𝒙 (𝟎. 𝟕) ^{𝟐𝟎−𝒙} ; 𝒙 = 𝟎, 𝟏, 𝟐, … , 𝟐𝟎

a) 𝑃(𝑋 ≥ 10) = 1 − 𝑃(𝑋 < 9) = 0.048

b) 𝑃(𝑋 ≤ 4) = 𝑓(1) + 𝑓(2) + 𝑓(3) + 𝑓(4) = 0.2375

c) 𝑃(𝑋 = 5) = 0.1789 ≈ 18% , NO 

Tha 30% is an incorrect percentage of operator errors. It's just that in this sample, the number of actual errors was less than the number of expected errors (which is 30% x 20, or 6 errors).