Punting - why value is the key

There was a discussion on the Grandouet thread about the fundamental aspect of betting, that was off topic and I think a few posters get annoyed when threads are derailed into side shows. Therefore I decided to start a thread of it's own. First up I'll have a stab at explaining why you cannot make money in the long run unless you bet at prices that under estimate the chances of your selection winning.

*apologises if I go a bit abstract and/or into a bit too much detail. Feel free to question anything I write

First a bit of house keeping

I will use the shorthand P(X) to represent the probability of event X happening eg if I toss a coin P(H) = the probability the coin lands on heads

For any numerical examples I will use the example of tossing a "fair" coin. By "fair" I mean P(H) = P(T) = 50%. For the more pedantic we can either assume that the probability of it landing on it's side is zero, or if you like, given that it lands on it's side all bets are void with stakes returned

For illustration of some of the points I will assume I have found a bookmaker that is betting on the coin toss but has incorrectly priced his market. I assume the following;

1. He bets 4/6 Tails.
2. He bets 11/10 Heads

Lastly I assume that each coin toss is independent of the last, i.e. knowing the result of the last coin toss gives me no further information about the outcome of the next coin toss

Concept 1: Expected Value

The expected value (EV) represents the weighted average of all possible values that a variable can take

If we take the coin toss market our friendly bookmaker has priced. If I place a €1 bet on heads at 11/10 my cashflow is as follows;

1. I will win €1.10 with probability 50%
2. I will lose €1 with probability 50%

From our definition of expected value we see this bet has positive expected value, or +EV. I will call this a "good" bet

+1.1*0.5 - 1*0.5 = 0.05

This means that if I place a large number of bets at 11/10 I will can expect to win 50% of my bets and lose the other 50%. However the profit on the winning bets more than cancel the losses on the the losing bets. So I play this market over and over again I should come out in front to the tune of 5% of my turnover.

If I place a €1 bet on tails at 4/6 my cashflow is as follows;

1. I will win €0.67 with probability 50%
2. I will lose €1 with probability 50%

From our definition of expected value we see this bet has negative expected value, or -EV. I will call this a "bad" bet

+0.67*0.5 - 1*0.5 = -0.1667

This means that if I place a large number of bets at 4/6 I can still expect to win 50% of my bets and lose the other 50%. However the profit on the winning bets is not enough cancel the losses on the the losing bets. So I play this market over and over again I should lose, to tune of 16.67% of my turnover.

So we should only make "good" bets

Concept 2: Variance

So we should only place "good" bets. However can we still lose money? Unfortunately the answer is Yes. Shrewd punters can still lose money over a finite series of bets and poor gamblers can can still win money over a finite series of bets due to variance

If we place only 1 bet we will lose money if tails comes up and will win if heads comes up. So our actual P&L will be either +€1.10 or -€1. Neither is equal to the EV mentioned above because of statistical variation or variance. So we win or lose money with probability 50%. Doesn't sound as good now does it?

However lets assume we place 10 bets on this market. We will win money if 5 or more heads come up. We will lose money if 4 or less heads come up. These probabilities can be evaluated using what is called a binomial distribution. I won't go into this but excel has a function to calculate these probabilities and there are numerous online calculators

If we place 10 bets we will win money with probability 62.3%. Better but not all that great you may think

If we place 100 bets we win if 48 or more coin tosses come up heads. Therefore the probability we win money (again from a binomial distribution) is 69.1%

If we place 100 bets we win if 477 or more coin tosses come up heads. Therefore the probability we win money (again from a binomial distribution) is 93.1%

What is happening is that the more "good" bets we place the higher the probability we win money overall. And also the more good bets we place the closer our profit on turnover gets to our EV

This leads us onto the fundamental mathematical theorem underlying this explanation.

Concept 3: The Law Of Large Numbers

The law of large numbers says, in lay mans terms, that the larger the number of "good" bets we place the closer and closer our profit on turnover will get to our EV (5% in the coin toss example)

If we could place an infinite number of bets we would be certain to achieve our EV (as a % of turnover), no more no less. i.e. in the coin toss example if we could place an infinite number of bets at 11/10 the probability we would win 5% on turnover is 1.

A simulated example

I simulated 160,000 coin tosses in excel (using the Rand() function) and calculated the profit on turnover. The attached graph shows that our profit on turnover (the purple/blue line) gets closer and closer to 5% (the grey line) as the number of coin tosses, and bets get larger. In this case we actually have a bad run of luck over the first quarter or so tosses but these losses even out over the long run. If we could simulate an infinite number of coin tosses the law of large numbers ensures that we would achieve a profit of 5% on turnover

FWIW I got 50.03% simulated tails and 49.97% simulated heads

Horse races aren't coin tosses!

That's all well and good, in the situation described above we know explicitly P(H) and P(T) before we place our bets. This is not the case we horse racing. Before a race we can never know the probability a horse will win a race with more than 1 runner with certainty. This is where knowledge of form, going, distances etc come in. We make an implicit estimation of the chances our horse wins, if the bookmaker offers better odds we think we should be getting we bet. There are a few metrics you can use to keep track of your ability top price up the horses you are backing. I keep track of P&L, Return on turnover, price taken v Betfair SP etc etc

I calculated the probabilities above very quickly so may have made a few small numerical error but the underlying theory is sound. Any queries fire away!

*if anyone is interested, for the coin toss example since the set up and probabilities are the same for each coin toss the weak law of large numbers suffices. In the case of betting on sports where each event and probabilities is different we must apply a more complicated form called Kolmogorov's Strong Law of Large Numbers