SOLUTION: . A company has three machines A, B and C which all produce the same two parts, X and Y. of all the parts produced, machine A produces 60%, machine B produces 30%, and machine C pr

Algebra ->  Probability-and-statistics -> SOLUTION: . A company has three machines A, B and C which all produce the same two parts, X and Y. of all the parts produced, machine A produces 60%, machine B produces 30%, and machine C pr      Log On


   

Question 1115761: . A company has three machines A, B and C which all produce the same two parts, X and Y. of all the parts produced, machine A produces 60%, machine B produces 30%, and machine C produces the rest. 40% of the parts made by machine A are part X, 50% of the parts made by machine B are part X, and 70% of the parts made by machine C are part X. A part produced by this company is randomly sampled and is determined to be an X part. With the knowledge that it is an X part, find the probabilities that the part came from machine A, B or C.
Answer by ikleyn(52767) About Me  (Show Source):
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Let aX = the number of parts X produced by machine A;

    aY = the number of parts Y produced by machine A;

    bX = the number of parts X produced by machine B;

    bY = the number of parts Y produced by machine B;

    cX = the number of parts X produced by machine C;

    cY = the number of parts Y produced by machine C.


Let P = aX + aY + bX + bY + cX + cY is the FULL NUMBER of all produced Parts.   


Then the probability that the part X came from machine A  is equal to  %28aX%29%2F%28aX+%2B+bX+%2B+cX%29,   and we have


    aX = (0.6*P)*0.4 = 0.24*P

    bX = (0.3*P)*0.5 = 0.15*P

    cX = (0.1*P)*0.7 = 0.07*P.


Therefore,  the probability that the part X came from machine A = %28aX%29%2F%28aX+%2B+bX+%2B+cX%29 = %280.24%2AP%29%2F%280.24%2AP+%2B+0.15%2AP+%2B+0.07P%29 = 0.24%2F%280.24%2B0.15%2B0.07%29 = 0.5217 = 52.17%.

Answer.   The probability that the part X came from machine A = 0.5217 = 52.17%.