Answer to Question #335065 in Statistics and Probability for mmoloki

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? [10] (b) What is the probability that no more than 4 out of 20 such failures are due to operator error? [10]

Answer 1

Let $X=$ number of failures due to operator error:$X\sim Bin(n, p).$

$p=0.3, q=1-p=0.7, n=20.$

(a)

$$P(X\ge 10)=1-(P(X=0)+P(X=1)$$

$$+P(X=2)+P(X=3)+P(X=4)$$

$$+P(X=5)+P(X=6)+P(X=7)$$

$$+P(X=8)+P(X=9))=0.047962$$

(b)

$$P(X\le 4)=P(X=0)+P(X=1)+P(X=2)$$

$$+P(X=3)+P(X=4)=0.237508$$