In the fabrication of steel beams, two types of flaws may occur: (1) the inclusion of...

  

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In the fabrication of steel beams, two types of flaws may occur: (1) the inclusion of a small quantity of foreign matter ("slag"): and (2) the existence of microscopic cracks. It has been found by careful laboratory investigation that for a certain size I-beam from a given fabricator, the mean distance between mi- croscopic cracks is 40 ft along the beam, whereas the slag inclu- sions exist with an average rate of 4 per 100 ft of beam. Each of these types of flaw follows a Poisson process. (a) For a 20-ft I-beam of this size from this fabricator, what is the chance of finding exactly two microscopic cracks in the beam? (Ans. 0.076) (b) For the same 20-ft beam, what is the chance of finding one or more slag inclusions? (Ans. O.551) (c) If a 20-ft beam contained more than two flaws, it would be re- jected. What is the probability that a 20-ft beam will be rejected? (Ans. 0.143) (d) Four 20-ft I-beams were supplied to a contractor by this fab- ricator last year. Assume that the flaw conditions among the four beams are statistically independent. What is the probability that only one of the beams had been rejected? (Ans. O.360) In the fabrication of steel beams, two types of flaws may occur: (1) the inclusion of a small quantity of foreign matter ("slag"): and (2) the existence of microscopic cracks. It has been found by careful laboratory investigation that for a certain size I-beam from a given fabricator, the mean distance between mi- croscopic cracks is 40 ft along the beam, whereas the slag inclu- sions exist with an average rate of 4 per 100 ft of beam. Each of these types of flaw follows a Poisson process. (a) For a 20-ft I-beam of this size from this fabricator, what is the chance of finding exactly two microscopic cracks in the beam? (Ans. 0.076) (b) For the same 20-ft beam, what is the chance of finding one or more slag inclusions? (Ans. O.551) (c) If a 20-ft beam contained more than two flaws, it would be re- jected. What is the probability that a 20-ft beam will be rejected? (Ans. 0.143) (d) Four 20-ft I-beams were supplied to a contractor by this fab- ricator last year. Assume that the flaw conditions among the four beams are statistically independent. What is the probability that only one of the beams had been rejected? (Ans. O.360)