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Grand National | The Maths of Why Betting is a Fool's Game

Updated: Aug 5, 2021

How do Bookies make their money? We look at the Maths of betting, and why to avoid it...

Today is Grand National 2019, so you may be thinking about having a flutter on a horse with a charming name. And though this might seem like harmless fun, I know as a Maths tutor that betting is an uneconomical use of money. In this post I will share with you the mathematical basis on which betting shops work.

The first thing to note is that betting houses do not win their money by gambling: they win it through other people gambling. This is counter-intuitive. The customer naturally thinks, “I place a bet on a horse. If the horse wins, I come out on top, and if it loses, the betting shop comes out on top. So we’re both gambling!” In reality, however, the betting shop wins regardless of the outcome of the race. For this reason, only the customers are truly gambling their money — some will win, some will lose. The same cannot be said for the bookies.

Before we consider the Grand National itself, a race between 45 different horses (which is a little complex), let’s start by considering a much simpler matter of probability — a coin toss.

On a fair coin, the chance of getting heads is 1/2 and the chance of getting tales is also 1/2. Fair odds for such results are what’s called evens, or 1/1. This means that your winnings are equal to your initial stake. So if you bet a pound on Heads, and the result comes up in your favour, you expect to get your initial stake back and a further pound on top.

Now let’s think about what would happen if a betting shop offered evens on the toss of a coin, and 200 people took the bet. Because there’s no reason to pick Heads over Tails (or vice-versa) we might expect 100 people to bet on Heads, and 100 people to bet on Tales. Let’s say each person bets £1. The betting shop first takes in £200, then they flip the coin. If it’s heads they have to pay £200 to the people who bet Heads (£100 in returning stakes, and £100 in winnings), and if it’s Tails they have to pay £200 to the people who bet Tails. So no matter what the result is, they’ll have to pay out all the money they initially took in. This means they make £0 profit. Not much of a business!

For this reason, no betting shop would give odds of evens on a coin toss. (Or if they did, they wouldn’t stay in business long!) Instead, they might offer odds of 4/5. This means that if you bet £5 (the number to the right of the slash) you win £4 (the number to the left of the slash) back on top of your initial stake. If you lose, you still lose your whole £5.

Let’s consider what happens in this scenario, assuming that the same number of people take the bet and each bet £1. 100 people lose their pound, and 100 win. Those who win get 80p each (which is 4/5 of £1). So the company wins £100, and lose £80 (80p x 100).This means the company gets £20 profit, whether the coin comes up Heads or Tails. This already seems like a better business model. But it’s a little more complicated than that.

What if the company offers 4/5 odds on Heads and 4/5 odds on Tails, but for some reason no one bets Heads? Perhaps there’s a rumour going around that Tails is more likely to come up. So 200 people bet on Tails. Now if the coin comes up Heads, the company makes £200, but if it comes up Tails, they lose 200 x 80p = £160.

Betting companies never want to be in a position where they stand to lose money, even if the odds are still in their favour (in this scenario, they still stand to win more than the stand to lose, but luck is a dangerous game to play.) So, to ensure that the company comes out on top no matter what, they change the odds. They say: from now on odds on Tails are 3/5 (meaning that if you bet a pound, you can win 60p) and the odds on Heads are 9/10 (so if you bet a pound, you can win 90p back.) The idea is to encourage people to bet on Heads. As Heads is just as likely to win as Tales, and now it is a more financially rewarding bet, new betters are much more likely to bet Heads. So let’s say a further 200 people bet on heads. The company has now taken in £400. If Tales wins, they pay out £200 (in stakes) + £160 (in profit) and so come up £40 ahead. If Heads win, they pay out £200 (in stakes) and £180 (in profit). So they still come out £20 ahead!

Now we have established a couple of principles. Firstly, the companies set the odds in such a way that no matter what happens, they come out on top. Secondly, they alter the odds to protect against dangerous betting patterns that expose them to risk. (For the same reason, companies will put a limit on the maximum amount of money an individual can bet. It doesn’t actually matter how many people bet a certain way, what counts is how much money is riding on a given result.)

Let’s see how this applies to the runners in the Grand National. There will be 45 horses in the race altogether. This means, if each horse has an equal chance of winning (which in reality they don’t), fair odds on any given horse would be 45/1, meaning if you bet £1 on the winner, you get £45 back. Looking on a couple of gambling websites, only 7 of the 45 horses are being given odds that would pay out more than 45 — the other 38 horses are given considerably shorter odds. Put another way, this means the betting shops are saying 38 of the 47 horses have a better than average chance of winning!

Here are the full list of the horses with their odds. I have ranked them from favourite (most expected to win) to least favourite (least expected to win):

TIGER ROLL 7/2

ANIBALE FLY 10/1

RATHVINDEN10/1

VINTAGE CLOUDS 12/1

MS PARFOIS 16/1

PLEASANT COMPANY 20/1

STEP BACK 20/1

MALL DINI 20/1

ROCK THE KASBAH 20/1

WALK IN THE MILL 22/1

JURY DUTY 22/1

PAIROFBROWNEYES 25/1

ONE FOR ARTHUR 25/1

RAMSEES DE TEILLE 25/1

MINELLA ROCCO 25/1

NOBEL ENDEAVOUR 25/1

UP FOR REVIEW 25/1

JOE FARRELL 25/1

SHATTERED LOVE 25/1

DOUNIKOS 28/1

ABOLITIONIST 33/1

GENERAL PRINCIPLE 33/1

BLAKLION 33/1

GO CONQUER 33/1

ALPHA DES OBEAUX 33/1

BAIE DES ILES 33/1

WARRIORS TALE 33/1

BALLYOPTIC 33/1

OUT SAM 33/1

THE YOUNG MASTER 33/1

VIEUX LION ROUGE 33/1

FOLSOM BLUE 33/1

VALTOR 33/1

CAROLE’s DESTRIER 40/1

YALA ANKI 40/1

CAPTAIN RED BEARD 40/1

TEA FOR TWO 40/1

MAGIC OF LIGHT 50/1

COGRY 50/1

ULTRA GOLD 50/1

SINGLEFARMPAYMENT 50/1

MILANS BAR 50/1

IMPULSIVE STAR 50/1

MONBEG NOTORIOUS 50/1

Remember: The number on the right of the slash is the gambler's stake; the left is their winnings. They ALWAYS get their stake back if they win, on top of their winnings. So if they put £2 on Tiger Roll and he won, they’d make a £7 profit. The same stake on Vieux Lion Rouge would win them £66 (2 x 33 / 1).

Let’s imagine that 45 people place a bet on the race. Let’s say everyone bets £10, and everyone bets on a different horse. The company takes in £450 (45 x 10) in stakes. As mentioned before, of the 45 runners, 38 would pay out more than 45 times what the better put in. So unless one of the 7 lowest ranked horses won, (Magic of Light, Ultra Gold etc.), the company would pay out less than it took in. In this scenario, the company would be hoping that Tiger Roll won. If he did, the gambler would only win £35 (7/2 x 10), and so they would take a total profit £405 (£450 - £10 stake on Tiger Roll, - £35 winnings on Tiger Roll.). If Valtor won, however, the company would make a considerably small profit: £110 (£450 - £10 stake on Valtor, - £330 winnings on Valtor). If, however, Milans Bar won, the company would make a loss! (£450 - £10 stake on Milans Bar, - £500 winnings on Milans Bar = -£60)

For this reason, you can be absolutely certain that there isn’t an equal amount of money riding on each horse, or the odds would look very different. Remember, when lots of people bet the same way, the company shortens the odds on that result to protect themselves, and lengthen the odds on unpopular bets. This means we know that many more people have been betting on Tiger Roll than on Milans Bar — hence the term “favourite.”

So now imagine that £100 has been bet on Tiger Roll, £50 has been bet on each of Anibale Fly and Rathvinden (the second and third favourite) and £10 has been bet on each of the other horses. The total of the stakes is £620 (£100 + £50 + £50 + 42 x £10). If the favourite wins, the company pays out £100 stakes + £350 winnings, but still keeps £170. If Anibale Fly or Rathvinda wins, the company pays out £50 stakes + £500, but still keeps £70. And if one of the outsiders wins, say Impulsive star, the company pays out £10 stake, £500 winnings, and keeps £110. The same is true for any of the other horses.

So if you are considering betting today, remember that wherever you place your money, you are essentially putting £1 on a coin toss in which you only stand to gain 80p.

Blog Post Crafted by Toby

Toby is one of our top Maths and English tutors, as well as our Chief Assessor, and he also works on our Admin Team. His proudest moment was when he tied Christopher Lee’s (AKA Saruman’s) bow-tie!

Toby is in charge of recruitment of new tutors. He conducts interviews with prospective tutors and assesses their lessons to get a feel for whether they have the teaching style we're looking for.