3) A plant producing silicon wafers for use in integrated circuits inspects each

Question

3) A plant producing silicon wafers for use in integrated circuits inspects each wafer prior to wafers are found to be defective and wafer defects occur Shipping. On average, 12% of the independently. Suppose that the plant hires one inspector. a) From when the inspector starts work in the mo rning, on average how many wafers would be inspected before finding one which is defective? b) What is the probability that the sixth wafer inspected will be the first to be defective? c) What is the probability that the tenth wafer inspected is the second to be defective? d) Wafers come to the inspector in lots of 10. What is the probability that at least two of the next four lots contain defective wafers? a nalantad everv hour. Typically, 1% of

Explanation / Answer

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials   
n = is the number of independent trials   
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 4 * 0.12
= 0.48
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 4 * 0.12 * 0.88
= 0.4224
III.
standard deviation = sqrt( variance ) = sqrt(0.4224)
=0.6499

PART D.
atleast 2 of the next 4 contains defective = P(X>=2)
P( X < 2) = P(X=1) + P(X=0)
= ( 4 1 ) * 0.12^1 * ( 1- 0.12 ) ^3 + ( 4 0 ) * 0.12^0 * ( 1- 0.12 ) ^4
= 0.9268
P( X > = 2 ) = 1 - P( X < 2) = 0.0732

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