Let's Take a Look at San Marino...

This Tuesday, San Marino lost in Warsaw to a Polish team that finished with a comfortable 5-0 victory. Coupled with the other group result of Montenegro 1-1 England, San Marino sits in a rather precarious position.

San Marino is not mathematically eliminated. But their chances are so slim it's truly remarkable.

Here is the table as it currently stands:

Team	Pld	W	D	L	GF	GA	GD	Pts
Montenegro	6	4	2	0	14	3	11	14
England	6	3	3	0	21	3	18	12
Poland	5	2	2	1	11	6	5	8
Ukraine	5	2	2	1	6	4	2	8
Moldova	6	1	1	4	3	10	-7	4
San Marino	6	0	0	6	0	29	-29	0

With four games left, it is mathematically possible for San Marino to still achieve the 12 points needed to qualify for the next round (recall that the top spot gets automatic qualification, and the top eight number two teams (of nine groups) play a head to head round to determine four more teams that qualify).

What is not immediately obvious, however, is just how hard that would be. Here are the remaining matches in this group:

June 7: Montenegro vs. Ukraine
June 7: Moldova vs. Poland
Sept 6: Ukraine vs. San Marino
Sept 6: England vs. Moldova
Sept 6: Poland vs. Montenegro
Sept 10: Ukraine vs. England
Sept 10: San Marino vs. Poland
Oct 11: Moldova vs. San Marino
Oct 11: Ukraine vs. Poland
Oct 11: England vs. Montenegro
Oct 15: England vs. Poland
Oct 15: San Marino vs. Ukraine
Oct 15: Montenegro vs. Moldova

Eight of these games have obvious results if we want San Marino to go on to the next round: England has to lose their last four, and San Marino has to win their last four. Plugging those results into the table, we get:

Team	Pld	W	D	L	Pts
Montenegro	7	5	2	0	17
San Marino	10	4	0	6	12
England	10	3	3	4	12
Poland	7	3	2	2	11
Ukraine	8	3	2	3	11
Moldova	8	2	1	5	7

Forcing England to lose all of their games brings Poland and Ukraine within one point of the 12 point best that San Marino can still mathematically achieve. And the five match-ups that haven't yet been accounted for are:

June 7: Montenegro vs. Ukraine
June 7: Moldova vs. Poland
Sept 6: Poland vs. Montenegro
Oct 11: Ukraine vs. Poland
Oct 15: Montenegro vs. Moldova

Since Ukraine and Poland play each other, that match must end in a draw, AND they have to outright lose the rest of their games. That gives us:

Team	Pld	W	D	L	Pts
Montenegro	9	7	2	0	23
San Marino	10	4	0	6	12
England	10	3	3	4	12
Poland	10	3	3	4	12
Ukraine	10	3	3	4	12
Moldova	9	3	1	5	10

And the last game left is Montenegro vs. Moldova. And Moldova can't win that one, or they leap frog San Marino as well.

That means of the thirteen remaining matches, all thirteen have to have specific results or else San Marino is eliminated. To top it all off, San Marino would have to overcome a 49 goal deficit to win the tie-breaker and make it to the qualification round.

Let's re-cap. In order for San Marino to qualify for the second round of UEFA qualifying, every single one of the following things has to happen:

1) June 7: Montenegro beats Ukraine, AND
2) June 7: Moldova beats Poland, AND
3) Sept 6: Ukraine loses to San Marino, AND
4) Sept 6: England loses to Moldova, AND
5) Sept 6: Poland loses to Montenegro, AND
6) Sept 10: Ukraine beats England, AND
7) Sept 10: San Marino beats Poland, AND
8) Oct 11: Moldova loses to San Marino, AND
9) Oct 11: Ukraine draws Poland, AND
10) Oct 11: England loses to Montenegro, AND
11) Oct 15: England loses to Poland, AND
12) Oct 15: San Marino beats Ukraine, AND
13) Oct 15: Montenegro does not lose to Moldova, AND
14) San Marino overcomes a goal deficit of 49 in four games, AND
15) San Marino finishes top 8 (of 9) amongst the second place finishers

And more to that last point, the current projected cut-off is 12.5 points, so even if San Marino were to finish second in their group, chances are they would finish 9th amongst the second place finishers.

Wow.

So, what are the odds of such a thing actually happening? Well, if I steal the current ratings from www.eloratings.net and plug them into my Elo rater (it's not a perfect fit, admittedly, I don't use the exact same model as eloratings.net does, but it'll work as a fairly good approximation), I can approximate the odds of specific results. The current Elo ratings are:

1924 England
1741 Ukraine
1659 Montenegro
1650 Poland
1351 Moldova
  860 San Marino

The calculated probabilities are in the following table, and the desired results are hi-lighted:

Home	Away	H-Elo	A-Elo	E[X]	W	D	L
Montenegro	Ukraine	1659	1741	0.51	0.3449	0.3332	0.3219
Moldova	Poland	1351	1650	0.23	0.1125	0.2368	0.6507
Ukraine	San Marino	1741	860	1.00	0.9938	0.0049	0.0013
England	Moldova	1924	1351	0.98	0.9644	0.0281	0.0075
Poland	Montenegro	1650	1659	0.61	0.4566	0.3159	0.2276
Ukraine	England	1741	1924	0.37	0.2140	0.3105	0.4755
San Marino	Poland	860	1650	0.02	0.0060	0.0229	0.9711
Moldova	San Marino	1351	860	0.97	0.9440	0.0439	0.0121
Ukraine	Poland	1741	1650	0.74	0.6107	0.2570	0.1323
England	Montenegro	1924	1659	0.89	0.8176	0.1354	0.0470
England	Poland	1924	1650	0.89	0.8254	0.1301	0.0445
San Marino	Ukraine	860	1741	0.01	0.0035	0.0138	0.9827
Montenegro	Moldova	1659	1351	0.91	0.8525	0.1112	0.0362

The the odds of all of these results going San Marino's way (which is simply ascertained by multiplying all the hi-lighted probabilities together) is 0.00000000000000000237823.

Or about 1 in 420 quadrillion.

And that's a very lower bound, conservative estimate, since we haven't even taken into consideration that San Marino has to beat each of their opponents by about twelve goals each.

A person has better chances at winning the Mega Millions lottery jackpot... twice.


So San Marino isn't mathematically eliminated yet, but no matter what odds your bookie gives you, don't take that bet.