(Update: thanks to Gazza White and the AFL subreddit for linking my post  it's already by far my most popular blog post!)
Towards the end of a sporting season, it's not unusual to hear the commentators call a team a "mathematical" chance to achieve some target  be that winning a premiership, making the finals, avoiding relegation, whatever. What this means is that there is at least one combination of events (usually discounting other teams being disqualified) that could bring it about, but it's almost vanishingly unlikely to occur.
Very seldom is this a more appropriate term than for the current chances of Greater Western Sydney getting into the top 8 and making the AFL finals this year  so much so that commentators probably aren't even aware that it
is a mathematical possibility.
Here is the current AFL ladder as of the end of Round 15 (courtesy of
FanFooty  note that the official AFL ladder is not actually up to date!)
Team 
P 
W 
D 
L 
For 
Agt 
Percent. 
Pts 

Hawthorn 
14 
12 
0 
2 
1645 
1167 
141 
48 
Geelong 
14 
12 
0 
2 
1556 
1216 
128 
48 
Essendon 
14 
11 
0 
3 
1483 
1142 
129.9 
44 
Sydney 
14 
10 
1 
3 
1379 
1048 
131.6 
42 
Fremantle 
14 
10 
1 
3 
1201 
954 
125.9 
42 
Richmond 
14 
9 
0 
5 
1387 
1190 
116.6 
36 
Collingwood 
14 
9 
0 
5 
1321 
1225 
107.8 
36 
Pt Adelaide 
14 
8 
0 
6 
1317 
1158 
113.7 
32 

West Coast 
14 
7 
0 
7 
1404 
1277 
109.9 
28 
North Melb. 
14 
6 
0 
8 
1435 
1210 
118.6 
24 
Carlton 
14 
6 
0 
8 
1331 
1219 
109.2 
24 
Adelaide 
14 
6 
0 
8 
1288 
1228 
104.9 
24 
Gold Coast 
14 
5 
0 
9 
1197 
1341 
89.26 
20 
Brisbane 
14 
5 
0 
9 
1133 
1451 
78.08 
20 
W. Bulldogs 
14 
4 
0 
10 
1102 
1433 
76.9 
16 
St Kilda 
14 
3 
0 
11 
1129 
1337 
84.44 
12 
Melbourne 
14 
2 
0 
12 
981 
1775 
55.26 
8 
W. Sydney 
14 
0 
0 
14 
1003 
1921 
52.21 
0 
In
green is our team of interest  Greater Western Sydney. They are currently winless at the bottom of the ladder, 8 wins behind the lowest top 8 side (Port Adelaide  in
red). Unfortunately for GWS, there are also 8 games left in the season, so one thing is immediately clear: GWS must win
all 8 of their games, and Port Adelaide lose
all 8 of theirs, for GWS to be any chance of making the finals (the two teams do not play each other, so this accounts for 16 separate games). If this happens, the ladder looks like this:
Hawthorn 
52 
Geelong 
52 
Fremantle 
46 
Essendon 
44 
Sydney 
42 
Richmond 
36 
Collingwood 
36 
Port Adelaide 
32 
GWS 
32 
West Coast 
28 
Carlton 
28 
Adelaide 
28 
North Melbourne 
24 
Gold Coast 
24 
Brisbane 
24 
Western Bulldogs 
16 
St Kilda 
16 
Melbourne 
8 
This on its own is still not enough to guarantee GWS a place, however  there are 9 other teams on the ladder that are also striving for a spot in the top 8. For GWS to make the finals,
none of these sides can finish with more than 32 points (8 wins) at the end of the season. Therefore every game that involves one of these sides  46 games, excluding the 16 already accounted for by GWS and Port's games  can make or break GWS's finals chances. In particular, West Coast, Carlton and Adelaide cannot get any more than 1 win for the rest of their remaining games. In fact, there are only 10 games that
don't affect GWS's chances  the games between top 7 sides, who already have more wins than GWS can possibly get and are guaranteed to place above them on the ladder.
Using a computer to calculate the possible combinations in which this could happen comes up with 150,744 ways for GWS to place equal 8th. Even assuming that all teams will have a 5050 chance of winning each game for the rest of the season (discounting draws), an assumption which is very kind to GWS to say the least, this would give them a 150,744 / 2
^{62} =
3.27 in a hundred thousand billion chance of finishing equal 8th on points.
To put this into perspective, imagine a lottery where you have to pick which 6 balls out of 40 will be drawn  a 1 in 3.8 million chance. Now imagine only entering that lottery twice in your life  and
winning both times. Even THAT would be twice as likely as GWS finishing
equal 8th, on a
good day.
Notice that I've mentioned GWS finishing
equal 8th. Even this herculean feat doesn't guarantee them a place  in the very
best case scenario of the 150,744, there will be 6 teams vying for 8th place on 32 points (on average in these scenarios, there will be 9.6). So GWS's bestcase scenario looks like this:


Geelong 
76 
Fremantle 
66 
Hawthorn 
64 
Essendon 
60 
Sydney 
54 
Collingwood 
52 
Richmond 
48 
Port Adelaide 
32 
Carlton 
32 
Adelaide 
32 
Gold Coast 
32 
Western Bulldogs 
32 
GWS 
32 
West Coast 
28 
North Melbourne 
28 
Brisbane 
28 
St Kilda 
28 
Melbourne 
28 
To make the finals, from this point they need to gain a higher percentage than the other 5 teams. Currently, they are on 52.21%, having scored only 1003 against their opponents' 1921 points. On the other hand, their currently bestplaced opposition, Port Adelaide, has a percentage of 113.7%, scoring 1317 to their opponents' 1158.
This informative site tells us that the average score in an AFL this season so far is 92.43, and the average margin for a game is 36.92. So a roughly "average" game of AFL would involve the winner with 110.89 points and the loser with 73.97. If we assume that GWS's 8 winning games follow this scoreline, as well as Port's 8 losing games, then we end up with GWS having an improved percentage of 75.22% and Port with a dented percentage, but still plenty enough for finals, of 93.33%.
So, obviously just winning is not going to be enough for GWS to leapfrog Port and its other finals rivals. Let's assume the same as above, but this time work on the assumption that GWS has somehow found a secret scoring weapon and is able to rack up ridiculous scores while keeping their opponents to an average score of 73.97. They would need to be able to score, on average, 167.78 points in order to beat Port's percentage  an average winning margin of 93.8 on their run home  and hope that none of their other rivals have had a similar lateseason percentage boost themselves. I'll leave it to someone else to work out how often a team has won 8 games in a row by an average margin of at least 93.8 in AFL history.
Our conclusion: is it possible for GWS to make the finals? Mathematically, yes. Are they going to make the finals? No. But it'd be a hell of a story if they did!