This preview shows page 1. Sign up to view the full content.
Unformatted text preview: OR360 Problem Set 1 Solutions to Problems 1 and 2
1. Use the definition of the "choose" notation and compute directly, starting from the righthand side. Bring the two fractions to a common denominator so they can be added, then simplify: n1 n1 + r1 r = = = (n  1)! (n  1)! + (r  1)!(n  r)! r!(n  r  1)! (n  1)!r + (n  1)!(n  r) r!(n  r)! n! n (n  1)!n = = r!(n  r)! r!(n  r)! r For a combinatorial argument, consider a particular element X in a group of n elements. A subset of r elements out of this group either contains X or it does not, hence the total number of subsets equals the sum of the number of subsets containing X and the number not containing X. In order to choose a subset containing X, we take X into the subset, and choose another r  1 elements out of the remaining n  1 elements in the group, so there are n1 ways to do r1 this. In order to choose a subset not containing X, we place X aside and choose the r elements out of the remaining n  1 elements, so there are n1 ways to do this. r 2. There are many different ways to prove this. The easiest to note is that if A, B, C are independent, then A is independent from any set formed with B, C (this is proved in Ross, p.78). This then implies that Ac is independent from any set formed by B, C by Prop. 3.8.1 (Ross, p.77), and thus in particular, Ac and B C c are independent. This can also be done without using the theorems in the book. Rewrite the left hand side of the equation as: LHS = P {Ac (B C c )} = P {Ac B C c } = P {B}  P {B A}  P {B C} + P {A B C} = P {B} (1  P {A}  P {C} + P {A} P {C} = P {B} (1  P {A})(1  P {C}) Meanwhile, the right hand side can be rewritten as: RHS = P {Ac } P {B C c } = (1  P {A})(P {B}  P {B C}) = (1  P {A})(P {B}  P {B} P {C}) = (1  P {A})P {B} (1  P {C}) . We see that the two sides are equal. (independence) (inclusionexclusion) (independence) = P {B}  P {B} P {A}  P {B} P {C} + P {A} P {B} P {C} 1 ...
View
Full
Document
This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell University (Engineering School).
 Summer '08
 WEBER

Click to edit the document details