# Probability question.

#### swinburne

Hi.

I have a question which I'm hoping someone in here might be able to answer, it would help resolve a debate at work.

The question is in relation to a type of sports sweepstakes that we organise at work, and whether there is an equal probabliity of finishing first or last.....

It revolves around The Premiership (English soccer league)......Here's how it works........

There are 20 teams in the league.

Each team plays each other twice, so the competition runs for 38 weeks.

There are 20 players in the sweepstake.

Each player puts in a fixed sum at the start of the season.

Every two weeks there is a draw and each player is randomly allocated a team for that 2 weeks.

The player is credited with the goals which that team scores during that time.

At the end of the season whoever has accumulated the most goals takes half the pot, and whoever has accumulated the least gets the other half.

********************

As stated above, the question is, is there an equal probability of finishing first or last??? Common sense tells most people "yes" but I guess you guys in here know all about that.

What I can say is that experience down the years has seemed to indicate that it's actually harder to finish last!! We've been doing this now for about a dozen years and it always seems to follow the same pattern; 1 or 2 people are out in front at the top end from about 2/3 of the way through the season, but the battle for last is always a huge scramble between 6 or so people.

I hope there is enough information above to allow the question to be answered. I have a suspicion that the fact that soccer is such a low scoring game (it's not at all unusual for a team to score 0 in a game) means there's more likely to be more people down at the bottom than the top, but I haven't got the maths to prove it!!!

Thanks.

Gareth. Dublin - Ireland.

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#### mXSCNT

Yes, if the total number of goals scored is low, then it's more likely that a person has close to 0 goals scored than a large number of goals scored, and hence more seeming "competition" for last place than first.

It's not "harder" though to come in last than to come in first--it's all just the luck of the draw, the chance of either being 1/20 (assuming ties are broken to leave one winner in each category).

#### CompuChip

Homework Helper
I suppose one could make some mathematical model, but the question is how accurate it would be. For example, to keep it tractable one would probably want to set the probability of any team winning from any other team to some fixed number (like 1/2). But even using a lot of historical statistics, the probability of a team winning depends strongly on the opponent, and lots of other factors which can hardly be taken into account.

Then there is indeed the problem of the goals... even the worst teams will have to play the worst teams some time, and the best teams the best teams.

#### swinburne

Thanks for the replys.

I think the two replies sum up the main viewpoints I've come across on this question........ 1) It's an equal chance (1/20) for everyone (to finish in ANY position in fact) end of story....... and 2) There are probably some other contributing factors that may change the balance of probability at one end of the table or the other in some way, but it's effectively impossible to put concrete numbers in.

I can't help feeling that point 1) above is just a statement after the fact, but perhaps that's "common sense" at work again from me!!!!

#### mXSCNT

Probability always depends on what you know. If you knew all those other variables, like exactly how good a particular team is against another particular team on a particular day, you'd know the outcome and there would be no chance involved. The only reason you ever use probability is the fact that you only have partial knowledge.

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