Accumulators: fun, but here's the maths

Accumulators have an obvious appeal: a small stake can turn into a big number. I like a well-built acca too. But the maths behind them is harsher than most people realise, and if you are going to bet them, you should know exactly what you are paying for the thrill.

How an accumulator pays

An accumulator links several selections into one bet. Every leg has to win. The odds are multiplied together, so the returns climb quickly.

Four legs at 1.90 each give combined odds of 1.90 × 1.90 × 1.90 × 1.90 = 13.03. A £2 stake would return £26.06. Four singles at 1.90 with £2 each would return £3.80 per winner. It is easy to see the appeal.

The catch is that the multiplication works on the bookmaker's margin as well as on the odds.

The margin compounds

Take a genuine 50/50 event. The fair price is 2.00. Suppose the bookmaker offers 1.90 on both sides. That two-way market adds up to 105.26%, and on either side your expected return is 0.50 × 1.90 = 0.95. You lose 5p for every £1 staked in the long run.

Now string four of those together. The true chance of all four winning is 0.5 × 0.5 × 0.5 × 0.5 = 6.25%, which is 1 in 16, so the fair acca price is 16.00. You are paid 13.03. Your expected return is 6.25% × 13.03 = 0.8145. You now lose about 18.5p per £1.

Each extra leg multiplies your expected return by another 0.95:

An eight-fold on coin flips at 1.90 costs you about a third of your stake on average. The price looks exciting because 169.84 is a big number. The fair price is 256.

A 5% margin on a single is a tax. The same margin on an eight-fold is a third of your stake.

A worked four-leg example with an edge

The table assumes no edge. The only reason we ever publish an acca is that we think each leg is individually overpriced. Here is how that plays out, using four legs where our probability beats the implied probability.

Combined odds: 1.80 × 1.95 × 2.10 × 1.70 = 12.53.

Our probability of all four winning: 0.58 × 0.53 × 0.50 × 0.61 = 9.38%.

Expected return: 9.38% × 12.53 = 1.175, or roughly +17.5% per unit staked. As singles, the legs are worth between +3.4% and +5.0% each. Combined, the edges compound in our favour, the same way the margin compounded against us in the table above.

So why not bet big accas all day? Two reasons.

First, you lose most of the time. Even with those edges, the acca loses 90.6% of the time. The chance of 20 of these in a row all losing is 0.906 to the power of 20, which is about 14%. One bettor in seven would go twenty accas without a single winner while doing everything right.

Second, our probabilities are estimates. If each of our numbers is two points too generous, the edges all but disappear, and if we are three points out, every leg is a losing bet and the compounding that looked so friendly turns straight back against us. Errors multiply just as edges do.

Why we cap at five legs and stake 0.25 units

Our acca rules are built around those two problems.

Never more than five legs

Every leg we add is another estimate that might be wrong, and another margin we have to beat. Suppose each leg is a genuine 52.5% shot at evens, a very healthy edge for a single. A four-fold wins 7.6% of the time. A five-fold wins 4.0% of the time, roughly 1 in 25. Beyond five legs, the win rate is so low that a sensible stake takes years to tell you whether the selections are any good. Five is where we draw the line.

A flat 0.25 units on every acca

One unit is 1% of your bank, so 0.25 units is a quarter of one percent. On a £1,000 bank that is £2.50, and the example above would return £31.33 if it landed.

Staying small is about the losing runs, not the maths of any single bet. Take that five-fold at 52.5% per leg. The chance of 50 in a row all losing is about 13%. At 0.25 units each, 50 losers cost 12.5 units, which is painful but survivable. At 2 units each, it would be 100 units: the whole bank.

No more than three a day

We publish a maximum of three accumulators a day. That keeps your maximum daily acca exposure at 0.75 units, and it stops us padding the card. If there aren't three accas worth publishing, there will be fewer.

If you build your own

• Only include legs you would bet as singles. If a leg has no edge on its own, it is dragging the whole bet down.
• Check the combined odds against the fair price. Convert each leg to a probability, multiply them together, and compare with the price you are offered.
• Don't add a short-priced "banker" to boost the odds. A leg at 1.20 adds very little return and still carries a margin.
• Keep the stake small and fixed. Do not increase it after a near miss. The next acca has exactly the same odds of landing.

If the price conversions above are new to you, our guide to decimal and fractional odds covers implied probability and overround step by step.

Every published acca is settled on the Results page alongside our singles, at the odds and stake we printed.