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Risk & Reward: Singles versus Accumulators

The simplest of all bets, whether it’s fixed odds, money line or handicap, is the single. With a single bet on a sports event, only one outcome is backed, and the bet can generally either win or lose, although as we saw in the last lesson other possibilities exists, particularly for Asian handicaps. With a simple win/lose single, a selection must be successful to achieve a return. And unless the odds for that selection are very long, the payoff for a win is sometimes not enough to attract many punters. Let’s look at how a punter can increase the size of his potential profit, and the consequence this has on the level of risk that he’s exposed to, by betting accumulators.

Suppose Liverpool to beat Manchester United, and Arsenal to beat Tottenham, are both fairly priced at 2.50. We could place two single £1 bets on each. If both lose we lose £2, if just one wins we make a profit of £0.50.  If both win we make a profit of £3.00. But what if we place one bet that required both teams to win? This time, if either team fails win our bet is lost. If both teams win, however, our profit will be £5.25 from a £1 stake (or an overall return of £6.25). How is this possible? Such a bet is called a double. The odds of this double are essentially calculated by multiplying the odds for the individual singles (in this example 2.5 x 2.5 = 6.25). Since the estimated probability of Liverpool winning is 40% and the estimated probability of Tottenham winning is also 40%, this means that the chance of both winning is 40% x 40% (or 0.4 x 0.4) which equals 16% (or 0.16). Calculating the inverse of 0.16 give us 6.25, the odds for this double.

Initially it might seem that the double represents a much better deal than placing the two singles separately. The profit over turnover for the winning double is 525% (£5.25 profit divided by £1 staked) whilst that for the two winning singles is just 150% (£3 profit divided by £2 staked). However, for the double there is only one way to make a profit – both selected teams must win – and the chance of that happening is only 16%. For the singles we still see a profit if only one teams wins, and the chance of that happening is much greater at 48%. [This is calculated by multiplying the probability of Liverpool winning, 0.4, by the probability of Arsenal not winning, 0.6 (=0.24 or 24%) and then doubling it, since there is also the option of Liverpool not winning and Arsenal winning, again 24%.] In fact if we also then add the probability of both teams winning, we find there is a 64% chance that we’ll see some sort of profit. Admittedly it won’t be as big as for the straight double, but it’s much more likely (in fact 4 times more likely) to happen. Essentially, what we are doing with the double is taking on more risk to gain a bigger reward compared to the singles. Risk and reward is a double-edged sword. You can’t have one without the other. As one increases so will the other. There is simply no such thing as a free lunch in gambling.

Assuming that the odds for Liverpool and Arsenal winning are fair we would naturally expect that over the long term, if we could bet these matches many hundreds or thousands of time, betting at such prices would neither make us a profit not lose us money. To help us understand why, we must introduce the idea of profit expectancy. Profit expectancy is merely a mathematical expression to specify what money we would expect to make, on average, from a particular betting outcome, and is simply the probability of the outcome multiplied by the profit for that outcome. The Liverpool/Arsenal double, for example, has a potential profit of £5.25 for a £1 stake. If we can expect to win that bet 16% of the time, the profit expectancy is £0.84. That is to say that on average we can expect to make a £0.84 profit from this winning double. However, we can also expect to lose our £1 stake 84% of the time, so our profit expectancy for this particular outcome is -£0.84. Combining both possible profit expectancies give an overall figure of £0.00, which is exactly what we would expect for fair odds. We can calculate profit expectancies for any bet and any possible outcome, which is what we have done below to compare the performance of the singles versus the doubles where the odds for both a Liverpool and Arsenal win are 2.50. To ensure we are comparing like with like, we must standardise the total turnover, in other words the total amount staked. If our double involves a £1 stake, the total staked on the singles must also be £1, or £0.50 on each.

Odds = 2.50 Profit Expectancy
Outcome Probability Singles Doubles
Liverpool and Arsenal win 16% £0.24 £0.84
Liverpool wins, Arsenal fails to win 24% £0.06 -£0.24
Liverpool fails to win, Arsenal wins 24% £0.06 -£0.24
Liverpool and Arsenal fail to win 36% -£0.36 -£0.36
All possible outcomes 100% £0.00 £0.00


Tabulating the profit expectancies like this allows us to easily identify where the risk and reward lies. In the long run it doesn’t matter whether we bet singles or doubles; at fair odds we’ll end up breaking even. The difference lies in the size and number of payouts along the way. Betting singles there will be more profitable payouts, but they will be smaller. Betting doubles there is just one large payout but it won’t happen very often.

What happens, however, if the odds we are betting are unfair? Suppose for example, our bookmaker decided he’s going to build in a 125% overround into his prices. Of course that would be extreme for a 1X2 bet but we’re using this to illustrate the effect it will have on the double, as compared to the singles. This time Liverpool and Arsenal, still with a fair expectancy of 40% each of winning, are available at odds of 2.00. The next table shows the profit expectancies for both the singles and the double.

Odds = 2.00 Profit Expectancy
Outcome Probability Singles Doubles
Liverpool and Arsenal win 16% £0.16 £0.48
Liverpool wins, Arsenal fails to win 24% £0.00 -£0.24
Liverpool fails to win, Arsenal wins 24% £0.00 -£0.24
Liverpool and Arsenal fail to win 36% -£0.36 -£0.36
All possible outcomes 100% -£0.20 -£0.36


Betting singles with an overround of 125% means we can expect to lose 20 pence for every £1 staked, or a profit over turnover of -20%.  Compare that figure to the equivalent for betting doubles, a whopping -36%. In effect we have compounded the disadvantage in the odds. Betting singles, the overround we face is 125% (or 1.25). Betting doubles it’s 156.25% or 125% x 125% (1.25 x 1.25 = 1.5625). The relationship between the overround and the punter’s overall profit expectancy is given by the following expression:

Profit expectancy = (1/overround) – 1, where the overround is expressed as a decimal.

In this example, betting singles means an overround of 1.25, and hence a profit expectancy of -0.2, since 1/1.25 – 1 = -0.2. For every 1 unit wagered we should expect to lose 0.20 units on average. Betting doubles, however, compounds the overround to 1.5625, and hence our profit expectancy jumps to -0.36 (1/1.5625 – 1), nearly double that for the singles.

Let’s introduce another match into the scenario, for example Newcastle to beat Sunderland again at unfair odds of 2.00 with a 125% overround. The compounded overround for a treble bet (backing Liverpool, Arsenal and Newcastle all to win) would be 1.25 x 1.25 x 1.25 = 1.953125, and our profit expectancy -0.488. As we introduce more and more matches into an accumulator bet, the potential reward rises dramatically, but the chances of it happening diminish at the same rate, and if the odds for each part are unfair, our profit expectancy becomes worse and worse. Imagine a 10-fold with each part priced unfairly at 2.00 with a disadvantage of 1.25. The profit expectancy for such a bet would be -£0.89 (rounded to 2 decimal places). In other words for every £1 we might wager on such bets we would on average expect to get back just 11 pence. It should be abundantly clear by now that if you haven’t got the odds in your favour, betting ever increasingly larger accumulators in the pursuit of larger (but riskier) rewards is the fast track to financial ruin. Of course, the key to any profitable betting, whether it is singles, doubles or larger accumulators, is to move the odds in your favour, that is to say find an edge over the bookmaker. To see how this works, the next lesson introduces the concept of value betting.

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