So, how good are the English premiership top seven?

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oops: got the maths wrong first time – this is the corrected version

Last weekend was said to be one of the worst in the history of English bookmakers (I know, your heart bleeds) as for the first time ever all seven top teams in the English premiership won – so ensuring a lot of fixed odds accumulator payouts.

But just how big an upset was this “for the books”?

Well, the English premier league has been going since the summer of 1992, so this is its 21st season. In the first two seasons there were 22 teams, and since then there have been 20.

So the total game “weeks” have been:

1992/93 – 1994/95: 42 x  2 = 84
1995/96 – 2012/13: 38 x 18 = 684
2013/14 (to last week): 21

Giving a total of 789. So just one week out of 789 makes it sound like the top seven are not very reliable.

But, of course, this ignores the fact that – on most weeks – at least one of the top seven is playing another team in top seven (this is certainly happening this week with Chelsea playing Manchester United).

On any given week the chances of this NOT happening are (for a 20 team league):

\frac{13}{19} \times \frac{12}{17} \times \frac{11}{15} \times \frac{10}{13} \times \frac{9}{11} \times \frac{8}{9} \times \frac{7}{7} = 0.198

While for a 22 team league these odds are:

\frac{15}{21} \times \frac{14}{19} \times \frac{13}{17} \times \frac{12}{15} \times \frac{11}{13} \times \frac{10}{11} \times \frac{9}{9} = 0.248

So, in other words there have been only 84 x 0.198 + 705 x 0.248 = 191 weeks (on average – I’d have to look at the precise run of games this season to be clear) where this was even possible.

And how often should the top teams win? If they decided games on tossing of coins (and we ignore draws as they are ‘wins’ for the bookies) then the top seven would only all win 1 out of 128 weeks. The fact they managed it after 191 weeks might suggest they were worse than that, but we cannot be sure – it probably needs a bigger sample size.

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