Probability / Permutations & Combinations question

Neil3030

I am trying to figure out if the odds of 80/1 for Real Madrid to win the league constitutes reasonably good value. The two teams play each other on Saturday, so assuming Madrid win (which I have an inkling they will), Barcelona would be ahead by 13 points, with 12 games left to play.

Lets then assume Madrid win all 12 of their remaining fixtures, which is about 4% likely (average odds of 1.3, 1/1.3=.77, to the power of 12 = 0.042).

By my reckoning, this would leave Barcelona needing 24 points from their remaining 12 fixtures. (Madrid would have 91 points, Barcelona are currently on 68).

Collecting 24 points from 12 games requires either:
Win 8 games (other 4 results irrelevant)
Win 7, draw 3, (other 2 results irrelevant)
Win 6, draw 6

This is where I hit my wall. Could someone please advise me what formula or combinations of formulae I would need to use from this point on?

It's the combination... permutation side of things that's confusing me. Were it a simple 8 wins from 8, it would simply be (liklihood of winning)^8, but the situation is more complex because, they have 12 games to get these 8 wins (495 possible combinations) and the 8 wins isn't the only way that they can do it.

*I know there's a short answer to this question: "no, it's not good value because otherwise bookmakers wouldn't offer it!"

Yakuza

Assuming Madrid win all 12 games is not a particularly safe assumption, so you really need to take into account the performance of both teams. This is fairly tricky stuff (which is why the like of Paddy Power employ quants to work this stuff out).

Each week (assuming no more clasicos after Saturday), you have a joint event with 9 possible outcomes (Mad win/FCBwin, Mad lose / FCBwin, Mad draw / FCBwin etc etc).
By assuming probabilities for each team winning / losing / drawing, you can work out probabilities for the joint events. Each of these joint events has a different effect on the relative points difference (currently 13 in FCB's favour). For example Mad win and FCB draw is a +2 difference to the relative totals, or Mad lose / FCB lose will have a zero difference to the relative score).

I think that you would need to use the multinomial distribution (12 trials, 9 outcomes), you would need to identify the outcomes where MAD get 13 more points than FCB and sum up the various probabilities; if you get a value of more than 1/80 (0.0125) then go for it Having said that, even in my actuarial stats / probability courses, I've barely scratched the surface of multivariate distributions and wouldn't be too confident about trying to work out the answer in that fashion.

Alternatively, (what I'd be more inclined to do, personally) would be to write a program to simulate the above events, run it a few million times and base my decision on what the most likely outcome of those 12 trials is.

Of course, in the real world, teams don't have fixed probabilities of winning /losing / drawing so the any conclusions drawn from the above is only an estimate :0

Neil3030

Thanks for that, I'll tinker about with MATLAB and see what I can come up with.

The plan wasn't to assume Madrid would win all their games, by the way. The only assumption was that they would win the next (and last) Classico. I had intended on factoring in the liklihood that Madrid would win all 12, with the final probability that Barca wouldn't hit the 92 required points.

Cheers again. N.

Neil3030

OK, so not really knowing anything about multinomial distributions, I just ran a simulation of sorts, using my limited MATLAB knowhow.

Took the win/loss/draw liklihoods from the games to date in the league, correcting them to assume Real win on Saturday:

p(barw)=(22/26)
p(bard)=(2/26)
p(barl)=(2/26)
p(madw)=(17/26)
p(madd)=(4/26)
p(madl)=(5/26)

I generated a 1,000,000 row by 2 column matrix. Column 1 was Madrid, Column 2 was Barca.

Using the above proportions, I assigned each cell as either 1 for a win, 2 for a draw or 3 for a loss.

For each row, I then specified how the combined events would alter the difference between the teams (from Madrid's perspective)

1 & 1 = 0
1 & 2 = +2
1 & 3 = +3
2 & 1 = -2
2 & 2 = 0
2 & 3 = 1
3 & 1 = -3
3 & 2 = -1
3 & 3 = 0

I then randomly plucked out blocks of 12, and summed the resulting figure for the points change.

Using those win/loss/draw proportions, it doesn't look good. By the law of averages, Madrid will actually lose around 6 points in the run in.

Only 17 of the 1,000,000 blocks resulted in a points swing of 13 or more :eek: Odds should be about 60,000/1 in that case!!

I ran the simulation again, assuming Madrid go on a run and win all of their remaining 12 games, and Barcelona conform to their seasonal averages.

This results in a 1.63% liklihood that Barca will drop the required points, or approx 60/1.