Cup Double

With Charlton having already played Oxford at home then Huddersfield away in the opening rounds (for us) of the Capital One Cup this season, what are the ex ante chances (see Note 1) of the exact same thing happening in the FA Cup? 

 

Well if my maths is right, it can be estimated as follows:

 

- probability of Oxford reaching the Third Round: 0.25 (see Note 2)

 

- probability of Charlton drawing Oxford at home in the Third Round: 0.008 (see Note 3)

 

- probability of Huddersfield reaching the Fourth Round: 0.4 (see Note 4)

 

- probability of Charlton reaching the Fourth Round (having been drawn at home to Oxford): 0.73 (see Note 5)

 

- probability of Charlton drawing Huddersfield away in the Fourth Round: 0.016 (see Note 6)

 

Therefore a reasonable approximation of the probability of playing Oxford at home then Huddersfield away in the first two rounds of the FA Cup after it has already occurred in the Capital One Cup is:

 

0.25 x 0.008 x 0.4 x 0.73 x 0.016 = 0.00000934 or 107,065/1

 

To put this in perspective this is less probable than tossing a coin 16 times and getting consecutive heads, or less probable than selecting seven random cards from a shuffled pack and each card being from the same suit.

 

Indeed whilst the probabilities will obviously be different for any given three club combination, I wonder if it has ever happened before?

 

Notes

1.       ‘Ex ante’ in this context means the estimated probability before the First Round of the FA Cup (featuring Oxford) had been drawn.

2.       Given Oxford United are riding high in League Two, before the First Round draw is made (see Note 1), I estimate the probability of them winning their First and Second Round ties to be 50% respectively, and thus 50% x 50% = 25% or 0.25.

3.       There are 63 other teams in the FA Cup Third Round draw, and an equal chance of being drawn home or away to any of them ie. 1/(63 x 2) = 0.008.

4.       As a midtable Championship side, this is a simple estimate of Huddersfield’s ex ante probability of reaching the Fourth Round in any given season – given they were given a relatively straightforward tie (Grimsby away), their actual probability increased significantly but this is not relevant for the calculation which considers the position ex ante (see Note 1).

5.       Current bookmakers odds of approx 4/6 imply a 60% probability (ie. 6/10) that Charlton win the tie at the first time of asking, whilst draw odds of 14/5 imply a 26% probability (ie. 5/19) that the tie needs a replay.  If a replay is required, I estimate the odds that Charlton still win the tie from that position (possibly after penalties of course) to be 50%.  Thus the estimated probability that Charlton reach the Fourth Round is 60% + (50% x 26%) = 73% or 0.73.

6.       There are 31 other teams in the FA Cup Fourth Round draw, and an equal chance of being drawn home or away to any of them ie. 1/(31 x 2) = 0.016.