Random Number Generation

Given a function random(0,1) which randomly generates 0 or 1 with equal probabilities, extend its functionality to generate random numbers for an interval [a,b], i.e. implement random(a,b), for all 0 =< a < b. Prove that the strategy is uniformly random or would you like to put some conditions on a and b?


Another version: Given a random function random() which generates 0,1 but with uneven probabilities, create a random function newrandom() which generates 0,1 with even probabilities. Uneven probabilities mean if probability of getting 0 from random is p, that of 1 is (1-p) but p!=(1-p)!=1/2