
Probability Proof help?
I have absolutely no idea how to tackle this problem. Any additional hints would be amazing.
Let T = {1, . . . , n}, and consider the sample space Ω = P (T ) = {S ⊆ T } of all subsets of T . Let P be the uniform probability measure on Ω (so P(S ) = (1/2)^n for each subset S ).
(a) For any A, B ∈ Ω, show that P(A ⊆ B) = ( 3 /4 )^n .
(b) For any A, B ∈ Ω, show that P(A ∩ B = ∅) = ( 3/4 )^n .
[Hint: let N : Ω → {0, . . . , n} be the random variable N (S ) = #S for any subset S . Then the events {N = 0}, . . . , {N = n} form a partition of Ω. Use the law of total probability to evaluate the probabilities in (a) and (b).]