When I was in graduate school, I read a paper about prediction markets.
8 teams have advanced to the quarter-finals of UEFA Champions League. With 4 of them coming from England (Manchester United, Liverpool, Tottenham and Manchester City), what is the probability that no two English teams are drawn together? If that happens, it's possible to have an all-English semi-finals. There are no rules in this draw, each team can be drawn against any other opposition. Step 1: Total number of possible drawings There are two ways to calculate this. First, without loss of generality, pick any team say Manchester United to be the first team in the draw, meaning the first ball coming up in the draw is for Manchester United. There are 7 possible opponents, all equally likely. One of them will be drawn to face Manchester United. Next, there are 6 teams remaining and the process continues. The next team drawn has 5 possible opponents, and so on. Hence the total number of possible drawings is 7*5*3*1=105. Alternatively, this can be calculated with combinations. Init...