The Guessing Game

Ryan writes two distinct real numbers on two separate piece of papers and puts them into two envelopes. Ruth chooses one of the envelopes randomly and she must guess whether the other real number in the closed envelope is less than, or greater than, the one she picked. Most people would think that the probability is about $\frac12$. However, in this presentation, we will prove by using probability theory and real analysis, we can develop a much more accurate method of guessing.