As a Liverpool fan, I was watching the draw for the 5th round of the FA Cup on Sunday afternoon at around 5:30PM. What I wanted was Liverpool at home to some minor league team. What I got instead was away to the league leaders – Arsenal. What was amazing was that of the other 7 ties, one of them featured the other two top 4 Premier League sides. What are the chances of the top 4 teams being drawn against each other in the FA Cup???

Picture courtesy of the FA

Well in the Figure it Out team, we thought that we would not only tell you what the chance of that actually is, but we would also show how we Figure it out.
Firstly we are assuming that (as in this year) all of the top 4 teams have made it through to this round. The next thing is that we are not bothered about which of the top 4 play each other, or who is home or away. That simplifies things a little. Secondly it doesn’t matter which of the 8 matches they are drawn in. That simplifies things a lot.
So if we assume that one of the top 4 is drawn in the first match, what is the chance that another team from the top 4 is drawn to play them. Well as with all probability questions it is the number of times that something is true divided by the total number of times it is possible.

In this case there are 3 other top 4 teams that the first team can play, and there are, at that stage, 15 teams in the bowl. So the probability is 3/15, which can be simplified to 1/5.
What about the second draw? Well if we assume that the next team out is also one of the top 4 (as we are not bothered about the order) then we need the chance that it too will draw a top 4 team. This time there is only 1 team that can be picked, and 13 teams left in the bag, so the probability is 1/13.

For the top 4 teams to be playing each other in 2 of the ties we need both of these things to take place. In Probability when we talk about something AND something else, then we need to multiply the probabilities. So in this case the probability of two of the ties featuring all of the top 4 teams, the probability is 1/5 x 1/13 which is 1/65. This comes to around 1.54%.

So the answer is that we should expect to see the top 4 teams playing each other in the FA Cup around 1.54% of the time, or around once every 65 years!
Ah well. At least Liverpool will not have so many teams to beat to win the overall competition…