My core ‘belief’ surrounding gambling is that you cannot beat the maths behind your bets. No matter how fancy you get with your staking plan or whether you just ‘know’ that Man Utd will be a let down, unless you’re being paid more than the true odds then you will lose long term. Failure to understand this is why there are so many people who throw money away at bookmakers or loss chasing systems such as Martingale and yet never seem to grasp why they can never make money.
Expected Value of a Bet
Earning money from gambling is not about how right you are, but how much you win when you are right. If you place a bet at decimal odds of 1.01 but the actual probability of the bet winning is 98% then despite winning on average 50 times for every loss, you will still lose long term since you will only have earned back 50% of your stake to compensate you for that loss.
That is the key concept behind expected value – our winnings need to statistically compensate more than our losses in order to churn out a profit in the long run. Let’s take another example; we flip a coin and if it’s heads then we gain £1 and if it’s tails then we lose £1. We know that we have a 50% chance of either heads or tails happening over the long run. We can then say that our expected value (EV) is:
50%*£1 + 50%*-£1 = £0
In other words we expect to breakeven in the long run. But what if we were given £2 every time heads came up yet we still lose £1 if its tails? Everything else stays the same but we are being paid double every time we win:
50%*£2 + 50%*-£1 = +£1
Now we expect to earn £1 every game we play on average. In this case we would say we have an edge and this is the key to making sure we will eventually make a profit.
But you can have an edge and still lose…
The astute among you will have noticed the word ‘eventually’. The unfortunate reality is that variance plays a bigger a role in how much we win or lose the smaller our edge, whether negative or positive. It can be far greater than we can comprehend which is why The Gambler’s Fallacy is so common among gambling addicts. Consider these two graphs showing a hypothetical profit vs number of bets made:
At first glance this looks like it’s clearly a losing system given that after over 4000 bets we have lost 100 times our stake. However look what happened after the next 6000 bets:
Suddenly we’ve recouped our losses and made a profit. This was simply a simulation of the ‘coin flip’ scenario again, made using the Bet Simulator Tool. Despite our long term average being zero we still can see substantial deviations from this. Now let’s compare the same scenario but change the probability of a win to 47% and 53%:
In the above graph most of our trials show a clear down trend however there is still a chance that we can be in profit or breakeven for up to 2000 bets.
A few of these trials were breakeven or in negative equity for up to a thousand bets but overall most showed a clear up trend. This does however show that even with an edge akin to the House’s in roulette, there is still a substantial chance that you will need to wait at least a thousand bets to confirm this. But this is just for even money bets remember. Now let’s do a scenario where the payout is 9/1 but we have an 10.6% chance of winning. This has the same EV as the 53% probability, even money scenario however our chance of winning any given bet is about five times lower:
Unsurprisingly our profit average after 10000 trials is about 600 however notice how many trials were in breakeven/negative territory for several thousands bets. Interestingly the trials denoted in purple and green were breakeven for 4000 bets but ended up being joint 2nd best performers after 10000 bets. This just further illustrates the point of how patient you need to be when betting.
If you have any questions feel free to leave a comment below.