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.
- What is the alignment probability P(|G intsc S| >=
1), if S is selected by blind picking?
- 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?
- 1 - 3C2 /
3C2 = 0.378
- The probability that group A contains one or both top-2 designs
(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
= (0.7)(0.75)2 + (0.4)(2)(0.75)(0.25)
= 0.39375 + 0.15