III. HOME WORK PROBLEMS

9. GOAL SOFTENING (4)


There are two classes of designs in the design search space, class A and B with 5 members in each class. If more convenient, you can think of an urn containing 10 otherwise identical balls labelled A or B. There are five A's and five B's which you can see. We also know the probability that either the top 1 or the top 2 design being contained in class A is 3/4. Suppose the Good enough set G is defined as the top-2 designs, and we plan to select 2 balls, i.e., selected subset size |S| = 2.
  1. What is the alignment probability P(|G intsc S| >= 1), if S is selected by blind picking?

  2. If you want to maximize the alignment probability, how do you select your S (explain why), and what is the alignment probability of this way of picking S?

SOLUTION:
  1. 1 - 3C2 / 3C2 = 0.378
  2. The probability that group A contains one or both top-2 designs are

    (0.75)2 + 2(0.75)(0.25) = 0.5625 + 0.375 = 0.9375

    versus that of group B,

    (0.25)2 + 2(0.75)(0.25) = 0.0625 + 0.375 = 0.4375

    we should randomly pick from group A.

    The alignment probability is:

    (1 - 3C2 / 5C2) * P(top-2 all in A) + (1 - 4C2 / 5C2) * P(exactly 1 of top-2 in A)
    = (0.7)(0.75)2 + (0.4)(2)(0.75)(0.25)
    = 0.39375 + 0.15
    = 0.54375


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