Gears produced by a grinding process are categorized either as conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). Suppose that 80% of the gears produced are conforming, 15% are downgraded, and 5% are scrap. Ten gears are selected at random.

a. What is the probability that one or more is scrap?
b. What is the probability that eight or more are not scrap?
c. What is the probability that more than two are either downgraded or scrap?
d. What is the probability that exactly nine are either conforming or downgraded?


(a) Let be the number of the ten gears that are scrap. Then Bin(10, 0.05).



(b) Eight or more are not scrap if and only if two or fewer are scrap.



(c) The probability that a gear is either degraded or scrap is 0.15 + 0.05 = 0.20. Let be the number of gears that are either degraded or scrap. Then Bin(10, 0.2).



= 0.3222

(d) The probability that a gear is either conforming or degraded is 0.80 + 0.15 = 0.95. Let be the number of gears that are either conforming or degraded. Then Bin(10, 0.95).

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