help me solve this please Problems: I. An inspector is chec I. An inspector is c

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help me solve this please

Problems: I. An inspector is chec I. An inspector is checking pressure vessels. He is concemed about the number of joins in the vessel that show either small fissures or odd discoloration. The joint and marginal mass probability distributions are given in the table where X = the number ofjoins that show small fissures Y = the number ofjoins that show odd discolorations Number of Small Fissures Of Odd Y=0 | 0.21 | 0.03 | 0.09 | 0.11 | 0.03 Ya11-0.13.1 0.0110.041 0.06 0.02 Y2 0.06 0.01 0.03 0.04 0.02 Discolorations Y 3 0.05 0.01 0.02 0.0 0.01 What is the probability that a join will have 3 small fissures and 2 odd discolorations? What is the probability that a join will have at least 2 small fissures but less than 2 odd discolorations? What is the probability that Njoin will have no more than I small fissure? a) b) c) d) What is the probability that a join with 1 small fissure will have no odd discolorations? e) A join has no odd discolorations. What is the probability that it has I small fissure? f) Are the number of small fissures and the number of odd discolorations independent? Prove your answer

Explanation / Answer

a) To find the probability that a join will have 3 small fissures and 2 odd discolorations we go to the cell which is intersection of the column (X=3) and the row (Y=2), and this probability is 0.04.

b) The probability that a join will have at least 2 small fissures but less than 2 odd discolorations is given by the sum of the cells

which is 0.21+0.03+0.09+0.13+0.01+0.04 = 0.51.

c) The probability that a join will have no more than 1 small fissure is the sum of the cells for columns corresponding to (X=0) and (X=1), which is 0.21+0.13+0.06+0.05+0.03+0.01+0.01+0.01 = 0.51.

d) The probability that a join with 1 small fissure will have no odd discolorations is the cell that is intersection of the column X=1 and the row Y=0, and it is 0.03.

e) The probability of a join having no odd discolorations is 0.21+0.03+0.09+0.11+0.03 = 0.47. Now, the proabbility of a join having 1 small fissure given that it has no odd discoloration is 0.03/0.47 = 0.0638.

f) Now, P(X=4,Y=0) = 0.08*0.47 = 0.04 != 0.03, and hence we conclude that the number of small fissures and the number of odd discolorations are not independent.

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