Metal casting processes are notoriously slow and expensive A

Metal casting processes are notoriously slow and expensive. A common problem facing many older casting processes is flashing. In casting, liquid metal is shot into a mold and rapidly cooled. A flash commonly forms on the piece at the spot in the mold where the metal flows. Sanding corrects minor problems with flashing, but severe problems may require scrapping the part. A particular casting operation historically has scrapped 10% of its parts due to severe problems with flashing.

(i) Find the probability that the the first part scrapped is the tenth one made.

(ii) Find the probability that the fourth part scrapped is the twentieth made.

Solution

I.

Note that for a geometric distribution

P(x) = [p^(x - 1)] (1 - p)

here, p = 0.90, the probability that it is not scrapped

Thus,

P(10) = (0.90^9)(0.10) = 0.038742049 [ANSWER]

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Here, we need three of the first 19 to be scrapped, and the 4th scrapped is the 20th made.

We use binomial probability for the first 19.

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    19      
p = the probability of a success (scrapped) =    0.1      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3 scrapped) =    0.179557656

And, we multiply this to the probability that the 20th is scrapped, 0.1. Thus,

P(4th scrapped is the 20th) = 0.017955766 [ANSWER]