I've tried to look objectively at our chances by looking at the points per game of all the teams around us and how the teams have finished in previous seasons.
Looking at the results so far this season and extrapolating them for the whole season, we have:
Points PPG Predicted Total
Tottenham 53 1.89 72
Arsenal 52 1.86 71
Chelsea 49 1.75 67
In order to get 4th place if all three of them perform as they have been performing all season, we will have to win all our games, which would take us to 72 points.
However, Arsenal and Tottenham have been known to be bad finishers while Chelsea have traditionally been strong finishers. So, looking at the PPG from last season:
First 28 Last 10 Difference
Tottenham 1.71 1.4 -18%
Arsenal 2.04 1.1 -46%
Chelsea 1.82 2.0 +10%
If they continue this trend, Tottenham will have a PPG of 1.55 which would give them 16 points in the last 10 games. Arsenal will have a PPG of 1 which would give them 10 points, and Chelsea will have a PPG of 1.92 which would give them 19 points. The table would then look like this:
In this case, we need to win all but one of our games to get 4th place.
But Chelsea are nowhere near where they were last year, and with all the problems they have, we can expect them to drop more points. Also, it would be too much to expect Arsenal to collapse again like they did last season. Looking at the 2009-10 season, Arsenal had a PPG of 2.07 in the first 28 games, and 1.7 in the last 10 games which equates to an 18% drop. If they follow this more realistic trend, they will have a PPG of 1.52 which will give them 15 points in the last 10, and get them to a total of 67 points.
Even if all these events go our way, we still need to win 9 out of our last 10 games to get to 4th. It's extremely unlikely but teams have gone on such runs before.