They've got the whole title in their hands. Or have they?

Hello,
it's a common football punditry cliche to say that a team has got their fate in their hands if all they have to do to e.g. win the league is to win all their games. One recent example of this is Robbie Savage's piece from 13th April, written after Liverpool beat Man City.

"we saw why the title is now solely in their hands" he said.

Some critics, such as @MikeBenchcapon, dispute this approach as it ignores:
a) the fact that sometimes more than one team can win the league by winning all their games. Think about a cup final – there the above logic would suggest that the destination of the cup was solely in both teams' hands.
b) the teams playing the team in whose hands it supposedly is obviously also have some say in the result.

Useful, but inaccurate
Whilst I do recognise the usefulness of an expression that means "they win the league/stay up/etc. if they win all their games", the critique is valid that "in their hands" is somewhat inaccurate. To demonstrate in whose hands the title really is, I was going to do an analysis of all the teams still involved in potentially title-deciding matches to be played after Savage's article.

Noble, but unfortunate
Unfortunately, at the time of Savage writing, even Arsenal and Everton could still have won the league and only four teams, Spurs, Fulham, Stoke and Swansea, weren't involved in any matches which could possibly have had any bearing on the title. This large number of games remaining and indeed teams involved makes the calculations horrifically complicated – not difficult really but just very very many, which would make just setting up the spreadsheet take more time than I have.

So what I've done is I've worked out a couple of hypothetical but not unrealistic examples using a proposed model, just for interest really and I suppose to demonstrate just how spread the fate of a title can be across multiple hands.

Example №1: Two-horse race on final day of season
The first example situation sees the league table as follows:
Liverpool, played 37,  86 points, vastly superior goal difference.
Chelsea, played 37, 84 points, vastly inferior goal difference.
Man City, played 37, 82 points.

Liverpool play Stoke.
Chelsea play Hull.

It doesn't take long to work out that any draw will do for Liverpool whereas Chelsea need to win and hope Liverpool lose. Man City are out of it. Some would say the title is in Liverpool's hands, but to what extent is that true?

Breaking it down
Now, we have two games, each of which has three possible outcomes, a home win for the title-chaser, a draw and a defeat. This gives us a total of nine possibilities.

1. Liverpool win, Chelsea win
2. Liverpool win, Chelsea draw
3. Liverpool win, Chelsea lose
4. Liverpool draw, Chelsea win
5. Liverpool draw Chelsea draw
6. Liverpool draw, Chelsea lose
7. Liverpool lose, Chelsea win
8. Liverpool lose, Chelsea draw
9. Liverpool lose, Chelsea lose

So each of these results has a 1/9 contribution to the overall spread of the title across various hands.

The first six result possibilities (Liverpool win or draw) lead to  Liverpool winning the title and are down to Liverpool having made good use of their hands to such an extent that all other hands (particularly Chelsea's) are rendered irrelevant. So the amount of title in Liverpool's hands is:
6*(1/9)=6/9=2/3

Chelsea get the title in possibility 7, which is done by them winning and by Stoke beating Liverpool. This means that their hands share this 1/9, giving Chelsea's and Stoke's hands an 1/18 from possibility 7.

In possibilities 8 and 9, the Liverpool-Stoke result doesn't matter, and the title has been decided by Hull's salvaging a draw/landing a shock win. So their hands get both of these ninths.

To sum up, and converting to eighteenths to aid comparability, the title is in:

Liverpool's hands: 12/18
Hull's hands: 4/18
Stoke's hands: 1/18
Chelsea's hands: 1/18

Whilst Robbie Savage (I haven't asked him) might think it is a bit silly to say that Hull have more say in the title race than Chelsea, I think it makes complete sense given that they only need a draw or a win, whereas Chelsea need a win. I mean, each football match is contested by two teams with equal numbers of hands and an equal say in the outcome of the match. You could adjust for teams being better by saying that their hands are more secure, if you wanted to, but for now I'm weighting all teams' hands equally. 

The other example is a bit more complicated and I think I've made my main point here so I'll save the other example for a later post. 

Comments welcome as ever. Thanks.