One of the Worst Bets That You Could Ever Make?

I published this before the start of the season ... and I still state that this bet was one of the worst I've ever see.

With Chelsea (13/8) and Leicester (5,000-1) disrupting the expected form guides ... there is no way that anyone will get this right!

So the last time I classified in six classifications, based on betting odds to win the league, ranging from Excellent (1) to Poor (6). So if any team moves outside their classification areas, it is highly likely that the bet will not be won. As we can see, currently, Chelsea and Leicester have a score of 5, with Liverpool and Watford with scores of 2, and lots of 1 scores. Man City, Arsenal, Man Utd, Stoke, Sunderland, Norwich and Bournemouth are performing are expected:

Introduction

As you may know I love numbers and combinations, so this article is a bit of fun on the analysis of likelihood, and how you can estimate complex odds. I find it intriguing about how odds companies use various analytical techniques to create an edge on the market [here][here].

Ever felt the bookies and the lottery companies were tricking you a little ... well read on?

Good bet or not?

William Hill are promoting a ￡2 bet that will win you ￡50 million if you can predict the end league positions of the English Premier League. There's no charity involved, so does the ￡50 million prize really represent your odds? It seems quite an easy bet on the face of it, and well worth a punt. So, can we actually predict whether this is a good bet or not and whether you should put your money on the lottery or on predicting these league positions?

It is almost certain that the ￡50 million will not be won for various reasons:

• The odds are massive stacked against a winner, and there isn't enough people who would take the bet for it to create one winner. Here's a good bet ... I will bet that no-one will win the prize!
• There will be at least one team who will considerably improve on their expected performance, and another who will drop like a stone... that's one of the reasons that the results of matches cannot always be predicted ... otherwise we could just assess the squads by computer, and the bookies would go out of business. The prediction of who these teams will be and how much they will improve/reduce by is almost impossible to predict. To have just one guess at the movements gives a punter virtually no chance to get it right.

First assumption

Let's make the assumption that we have a division with two leagues and where only six teams are likely to be in the Top 6 (Man Utd, Chelsea, Liverpool, Spurs, Arsenal and Man City) - which is actually not a strong assumption. So we get:

6 x 5 x 4 x 3 x 2 x 1 = 720 combinations

So there are 720 ways that the Top 6 could end up.

Next, if we assume that the rest of the teams are equal, there are 14 teams who could finish in 7th position, 13 in 8th position, and so on, to give:

720 x 14 x 13 x 12... x 2 x 1

If we now calculate the number of permutations we get:

720x14! = 62,768,369,664,000

So if 62,768,369,664,000 bets (62 million million) were made with bet of ￡2, the income will be ￡125,536,739,328,000, and only one bet is likely to succeed. This will make a  profit of ￡125,536,689,328,000 (￡12.5 million million) ... making would make it one of the worst bets ever!

The chance of anyone predicting the whole of Premium League would then be as likely as ...

space aliens landing on the pitch in Manchester, while Manchester Utd were scoring a goal against Aston Villa with two minutes to go.

While, of course, the rest of the 14 left are not equal, the odds would still be massively against the punter. The assumption of the Top 6 is not even an exact one, as another team, such as Everton, could break through into the Top 6, and defeat our assumption of 720 combinations.

There is a good basis for the assumption of the six teams being the likely Top 6, as the last five years have produced:

1     Chelsea (20144/2105 Champions)
2     Manchester City
3     Arsenal     
4     Manchester United
5     Tottenham Hotspur
6     Liverpool     

1     Manchester City  (2013/2104 Champions)
2     Liverpool     
3     Chelsea     
4     Arsenal
5     Everton
6     Tottenham Hotspur

1     Manchester United (2012/2013 Champions)
2     Manchester City
3     Chelsea     
4     Arsenal     
5     Tottenham Hotspur
6     Everton

1     Manchester City (2011/2012 Champions)
2     Manchester United
3     Arsenal     
4     Tottenham Hotspur
5     Newcastle United
6     Chelsea

1     Manchester United  (2010/2011 Champions)
2     Chelsea     
3     Manchester City
4     Arsenal     
5     Tottenham Hotspur
6     Liverpool

So the Top 6 are fairly predictable, but the past few years have seen another team join the big hitting teams. The problem we have is to define the order of these times. While there are 720 combinations, it is likely that Chelsea will appear in the top two or three, so more likely combinations see Chelsea and Manchester City in the top two or three, and Liverpool and Spurs in the 5th and 6th place.

So, in this case, the ￡50million prize is thus a red herring, and is just there as a shiny attraction, as it could be ￡500million, and it would still would not reflect the odds. The prize will thus be given to the person who is the nearest to the final league positions - but only for ￡100K.

An improved guess ...

Let's now say there are three divisions within the league: Good teams; Average teams; and Weaker teams. In terms of the odds the leagues are then:

• Good teams: Chelsea (13/8); Man City (3/1); Arsenal (4-1);  Man Utd (5-1) and Liverpool (28-1);
• Average teams: Tottenham (150-1); Everton (300-1); and Southampton (300-1),
• Weaker teams: Crystal Palace (3,000-1); Newcastle (3,000-1); Stoke (3,000-1), Swansea (3,000-1), West Ham (3,000-1), Aston Villa (5,000), Leicester, Sunderland, West Brom (5,000-1), Bournemouth (7,5000-1), Norwich (7,500), Watford (10,000-1).

If we are extremely lucky, the teams will end up in these sub-divisions, we can estimate:

Possible combinations 5! x 3! x 12! =344,881,152,000

So, with the highly unlikely chance of the teams actually following into their divisions, we end up with 344,881,152,000-1 of getting it right.  Again ... not a good bet.

A final estimate ...

We are really struggling to make this bet make sense. Where did the ￡50 million prize come from? Finally, as a further estimate, the odds for the teams seem to split into the following finer divisions:

• Excellent team: Chelsea (13/8);
• Good teams: Man City (3/1); Arsenal (4-1);  Man Utd (5-1).
• Other good team: Liverpool (28-1);
• Average teams: Tottenham (150-1); Everton (300-1); and Southampton (300-1),
• Weaker teams: Crystal Palace (3,000-1); Newcastle (3,000-1); Stoke (3,000-1), Swansea (3,000-1), West Ham (3,000-1),
• Very weak teams: Aston Villa (5,000), Leicester, Sunderland, West Brom (5,000-1), Bournemouth (7,500-1), Norwich (7,500), Watford (10,000-1).

The odds for Chelsea coming first is 13/8, but let's just assume they will win the league, and that Liverpool will finish 5th. Now we get:

1 x 3! x 1 x 3! x 5! x 7! = 21,772,800-1

So if we assume the 13/8 on Chelsea will win, and that Liverpool will come 5th, we are actually at the point where the bet becomes  a bit more sensible.

Our last estimate is:

• Excellent team: Chelsea (13/8);
• Good teams: Man City (3/1); Arsenal (4-1);  Man Utd (5-1).
• Other good team: Liverpool (28-1);
• Average teams: Tottenham (150-1);
• Less Average teams: Everton (300-1); and Southampton (300-1),
• Weaker teams: Crystal Palace (3,000-1); Newcastle (3,000-1); Stoke (3,000-1), Swansea (3,000-1), West Ham (3,000-1),
• Very weak teams: Aston Villa (5,000), Leicester, Sunderland, West Brom (5,000-1),
• Extremely weak teams: Bournemouth (7,500-1), Norwich (7,500), Watford (10,000-1).

This makes the assumption of the teams falling into the divisions, and then that Chelsea win, Liverpool are 5th, and Spurs at 6th. This gives us:

1x 3! x 1 x 1 x 2! x 5! x 4! x 3! = 207,360-1

The odds of moving between the subdivision are now quite difficult to predict, and for Chelsea, Liverpool and Spurs to finish in their positions, so the previous estimate was probably the best we can do.

Your best bet?

So what is your best bet? Well don't take this bet! But if you are, you are not competing with anyone so the best bet is to go with the odds (unless you know something that the bookies don't know about ... which is unlikely):

1. Chelsea (13/8); 2. Man City (3/1) 3. Arsenal (4-1) 4. Man Utd (5-1). 5. Liverpool (28-1); 6. Tottenham (150-1); 7. Everton (300-1); 8. Southampton (300-1), 9. Crystal Palace (3,000-1); 10. Newcastle (3,000-1); 11. Stoke (3,000-1), 12. Swansea (3,000-1), 13. West Ham (3,000-1); 14. Aston Villa (5,000) 15. Leicester, 16. Sunderland, 17. West Brom (5,000-1), 18. Bournemouth (7,500-1), 19. Norwich (7,500), 20. Watford (10,000-1).

The bookies wouldn't like this, as you have the odds a bit more in your favour, so they want you to mix it up a bit. The more you mix it up, the higher your overall odds will be of getting it right. The 50million prize is likely to be modeled on the best case. Even if every person in the UK put their money on all the most likely end positions, it is unlikely that anyone will win. 

Most of the difficulty will be after Position 9, as there are similarly ranked odds. A particular challenge is around Position 9 and 13, and Position 14 and 17, along with Position 18 and 20.

If we assume that the bookies think that this is 25,000,000-1 (and that's a massive assumption). Let's take an alternative bet for the 2nd favouriate for each of the following (I've avoided the winner of the Scottish Premier League for obvious reasons):

• West Brom to be best Midlands Club (7/2). [Stoke fav]
• Man City to win league (3/1). [Chelsea fav]
• Liverpool to win FA Cup (8/1). [Arsenal fav]
• Wigan to win League 1 (6/1) [Sheff Utd fav]
• Hull to win Championship (10/1) [Derby fav]
• Luton to win League 2 (6/1) [Portsmouth fav]
• Arsenal to win League Cup (6/1) [Chelsea fav]
• Aberdeen to win Scottish Cup (6/1) [Celtic fav]
• Aberdeen to win Scottish League Cup (7/1) [Celtic fav]

This gives odds of 34,228,656-1. The major problem that we have with the prediction of getting all the positions correct, is that one error in a position will cause the bet to fail. So we tend to do an any x from y, so that if one or more fail, there is still options to win. Any 7 from 9 is calculated from the formula:

Combinations = 9! / (7! x (9-7)!) = 9 x 8 / (2) = 36

So we have 36 bets, but any of the seven can win and will give us a winning bet. This, at least, overcomes the problems of the all-or-nothing ￡50 million bet.

A better bet

So, if we assume that it is likely that the person will win ￡100K, with our first guess, what would be the equivalent bet? A Top 7 and 8 prediction (based on our assumption) will give:

10,080-1 (Top 7), 283,920-1 (Top 8)

So a much easier bet to win the ￡100K would be to predict the Top 8, as would bring a reward to ￡567,840 for a ￡2 stake. Here are the odds for these:

The numbers with E+11 represent 11 zeros on the number.

A simple example

You have 20 cards in a deck, and now you must be every one right. What is the chances for the first turnover ...

1 in 20

Well you did that ... well done! So now what's the chances that you have predict the first two ... well you have a 1-in-20 on the first card and if you got that one, then it 1-in-19 for the second one to give:

1-in-380 (20x19)

Soto get all 20 right, we get:

20x19x18...x3x2x1

In maths we define that as 20! or 2,432,902,008,176,640,000. So you have a:

1-in-2,432,902,008,176,640,000

chance of predicting all the cards in the deck. If you bet 1p for each go, the prize money would have to be: ￡24,329,020,081,766,400 for you to be worth your while.

Conclusions

I repeat .. there is virtually no chance of winning this! If you put ￡2 on this bet, you would be better to putting it on every home team to win this Saturday. No-one is likely to win this bet ... so William Hill have a ￡100K prize for the person who gets nearest. As long as there's 50,000 people putting on this bet, William Hill will make a profit.

This is an all-or-nothing bet, so if you get 19 correct (well that's not possible is it?) --- if you got 18 right, and two round the wrong way ... you get NOTHING!!!! -- well ￡100K which is not a good reward for such a good guess.

William Hill know that at least one team will not fit into their predicted rankings, and that is why they will win, as everyone else will go generally with the odds. So, I'll do an analysis later in the year, and see if we can find out how many teams have fitted into their leagues. So here is my starting point:

• Div 1: Chelsea (13/8);
• Div 2: Man City (3/1); Arsenal (4-1);  Man Utd (5-1).
• Div 3: Liverpool (28-1);
• Div 4: Tottenham (150-1); Everton (300-1); and Southampton (300-1),
• Div 5: Crystal Palace (3,000-1); Newcastle (3,000-1); Stoke (3,000-1), Swansea (3,000-1), West Ham (3,000-1),
• Div 6: Aston Villa (5,000), Leicester, Sunderland, West Brom (5,000-1), Bournemouth (7,500-1), Norwich (7,500), Watford (10,000-1).

Please check back soon, and I'll see if the odds were actually in your favour.

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If you are interested, here an overview of the advert:

Here is the text from the bet:

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