Suppose that an eddy current nondestructive evaluation technique for identifying

Question

Suppose that an eddy current nondestructive evaluation technique for identifying cracks in critical metal parts has a probability of around 0.10 of detecting a single crack of length .003 in. in a certain material. Suppose further that n = 8 specimens of this material, each containing a single crack of length .003 in., are inspected using this technique. Let W be the number of these cracks that are detected. Use an appropirate probability model and evaluate the following:

(a) P [W = 3]

(b) P [W <= 2]

(c) E W

(d) Var W

(e) the standard deviation of W

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
a.
P( X = 3 ) = ( 8 3 ) * ( 0.1^3) * ( 1 - 0.1 )^5
= 0.03307
b.
P( X < = 2) = P(X=2) + P(X=1) + P(X=0) +
= ( 8 2 ) * 0.1^2 * ( 1- 0.1 ) ^6 + ( 8 1 ) * 0.1^1 * ( 1- 0.1 ) ^7 + ( 8 0 ) * 0.1^0 * ( 1- 0.1 ) ^8 +   
= 0.96191
c.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 8 * 0.1
= 0.8
d.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 8 * 0.1 * 0.9
= 0.72
e.
standard deviation = sqrt( variance ) = sqrt(0.72
=0.84853

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