Components coming off an assembly line are either free of defects (S, for succes

Question

Components coming off an assembly line are either free of defects (S, for success) or defective (F, for failure). Suppose that 70% of all such components are defect-free. Components are independently selected and tested one by one. Let y denote the number of components that must be tested until a defect-free component is obtained.

(a) What is the smallest possible y value, and what experimental outcome gives this y value? What is the second smallest y value, and what outcome gives rise to it?
The smallest value of y is _____ . The outcome corresponding to this is ____ .
The second smallest value of y is _____ . The outcome that corresponds to this is ______ .

(b) Determine the probability of each of the five smallest y values. You should see a pattern that leads to a simple formula for P(y), the probability distribution of y.
P(y = 1) =  
P(y = 2) =   
P(y = 3) =   
P(y = 4) =  
P(y = 5) =

Explanation / Answer

we are given that probability of event is 0.7 , so probability of not event is 1- 0.7 = 0.3

smallest value of y is 1 , which is probability of defect is 0.7

second smallest value is 2 , which occurs when

P(y = 2)= P(not1) P(2) = 0.3 *0.7 = 0.21

P(y = 3) = P(not1) P(not2)P(3) = 0.3*0.3*0.7

P(y = 4) = P(not1) P(not2)P(not3)P(4) = 0.3*0.3*0.3*0.7

P(y = 5) =P(not1) P(not2)P(not3)P(not4)P(5) = 0.3*0.3*0.3*0.3*0.7

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