A machine produces articles and an average of 3 percent of these articles are de

Question

A machine produces articles and an average of 3 percent of these articles are defective. In a batch of 400 articles, calculate the probability that there are more than two defectives, (a) by using the binomial distribution. (b) by using the normal approximation to the binomial distribution. (c) Is the normal approximation in reasonable? Why yes? or why not? Each of 30 boxes in a line contains a single red marble, and for 1 lessthanorequalto k lessthanorequalto 30, the box in the k^th position also contains k white marbles. Tom begins at the first box and successively draws a single marble at random from each box, in order. He stops when he first draws a red marble. Let P(n) be the probability that Tom stops after drawing exactly n marbles. What is the smallest value of n for which P(n) lessthanorequalto 1/30?

Explanation / Answer

(A)
p = 0.03
P(X>2) = 1 - P(X<=2)
P(X<=2) = P(X=0) + P(X=1) + P(X=2)
P(X=0) = (1-p)^400 = 0.0000051
P(X=1) = 400C1 * p * (1-p)^399 = 0.000063
P(X=2) = 400C2 * p^2 * (1-p)^398 = 0.000390

P(X<=2) = 0.000459
P(X>2) = 1 - 0.000459 = 0.999541

(B)
mean = np = 0.03*400 = 12
std. dev. = sqrt(0.03*0.97*400) = 3.4117
P(X>2) = P(z > (2-12)/3.4117) = P(z > -2.9311) = 0.9983

(C)
Yes the normal approximation is resonable because population size is very large.

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