And, or, not: the rules of probability

The question

"I think the home side won't lose. How do I turn the home price and the draw price into one number?"

Knowing how to combine betting probabilities is the grammar of everything else in this Academy. Pricing a double chance, laying instead of backing, checking a Correct Score book, working out a double: each is one of three simple rules. This lesson gives you those rules and shows where each one breaks.

The idea in one sentence

Not is one minus the chance, or adds the chances and takes away any overlap, and and multiplies them, but only when one result tells you nothing about the other.

The picture

Think of every possible end to a match as a patch of ground, with the size of each patch equal to its probability. The whole field adds up to 100%. Match Odds splits the field into three patches that don't overlap: home, draw and away.

Question	Rule	Picture in words	Example
Not away?	1 − P(away)	Everything outside the away patch	1 − 25% = 75%
Home or draw?	P(home) + P(draw)	Two patches side by side	47.6% + 27.8% = 75.4%
Over 2.5 in match A or match B?	P(A) + P(B) − P(both)	Two patches that overlap; count the overlap once	55% + 50% − 27.5% = 77.5%
Both home sides win (different matches)?	P(A) × P(B)	A slice of a slice	50% × 40% = 20%

The first two rows need nothing but addition. The last two rows need you to know how the events relate to each other, and that's where most mistakes live.

Mutually exclusive: the easy "or"

Two outcomes are mutually exclusive when they can't both happen. Home, draw and away in one match are the classic case, and so are the scorelines in the Correct Score market. For these, "or" is plain addition.

Overlapping: the careful "or"

Over 2.5 in match A and over 2.5 in match B can both happen. If you add 55% and 50% you've counted the "both" case twice, which is how you end up with an impossible 105%. Take the overlap away once.

Independent: the only safe "and"

Two events are independent when knowing one tells you nothing about the other. Two home wins in different leagues on the same Saturday are close to independent. A home win and over 2.5 in the same match are not, and multiplying them gives the wrong answer (Lesson 3.4).

Worked Betfair example

Arsenal are at home in the Match Odds market. You think they won't lose and you want the best way to back that view. The back prices are home 2.10, draw 3.60, away 4.00, and the best lay price on the away side is 4.10. (Illustrative prices.)

1. Turn prices into chances. Implied probability is 1 ÷ odds: home 1 ÷ 2.10 = 47.62%, draw 1 ÷ 3.60 = 27.78%, away 1 ÷ 4.00 = 25.00%.
2. Check the book adds up. 47.62% + 27.78% + 25.00% = 100.40%. The extra 0.40% is the overround, tiny on Betfair.
3. Use the "or" rule. Home and draw can't both happen, so home-or-draw = 47.62% + 27.78% = 75.40%. The combined price is 1 ÷ 0.7540 = 1.326.
4. Build it by dutching £100. Split the money in proportion to each implied chance: £100 × 47.62 ÷ 75.40 = £63.16 on home and £100 × 27.78 ÷ 75.40 = £36.84 on the draw.
5. Check both outcomes pay the same. £63.16 × 2.10 = £132.63 and £36.84 × 3.60 = £132.63. Either way you get £132.63 back, a profit of £32.63.
6. Take off 2% commission. £32.63 × 0.98 = £31.98 net if Arsenal don't lose. If they lose, you lose the £100.
7. Now use the "not" rule instead. "Not away" is the same bet as laying the away side. Lay at 4.10 with £100 of liability: lay stake = £100 ÷ (4.10 − 1) = £32.26. If Arsenal don't lose you win £32.26, or £31.61 after commission.
8. Compare. Same £100 at risk, same outcome. Dutching pays £31.98, laying pays £31.61. Here the dutch is 37p better because the gap between the away back and lay price costs you a little.

Verdict: the two routes are the same bet written two ways, one with "or" and one with "not". Work out both and take whichever pays more on the day.

A quick "and": the double

Now two separate matches, each with a fair home-win chance of 50% and 40%. Nothing links them, so "both win" is 0.50 × 0.40 = 20%, fair odds 5.0. The chance that at least one wins is 1 − (0.50 × 0.60) = 70%. That last sum uses "not" and "and" together: at least one wins is "not (both lose)".

The formula

Not

$$P(\text{not } A) = 1 - P(A)$$

• P(A) is the probability that A happens.

In plain English: the chance something doesn't happen is everything that's left. On the exchange, laying A is betting on "not A".

Or, when the events can't both happen

$$P(A \text{ or } B) = P(A) + P(B)$$

• A and B are mutually exclusive outcomes, such as home and draw.

In plain English: side-by-side patches just add up. That's the maths behind dutching and double chance.

Or, when the events can overlap

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

• P(A and B) is the chance both happen, the overlap.

In plain English: add the two, then take the overlap off once so you don't count it twice.

And, when the events are independent

$$P(A \text{ and } B) = P(A) \times P(B)$$

• A and B are independent: knowing one tells you nothing about the other.

In plain English: multiply, but only for events that really are unconnected. For linked events you need conditional probability: P(A and B) = P(A) × P(B given A), covered in Lesson 3.2.

At least one

$$P(\text{at least one of } n) = 1 - \prod_{i=1}^{n} \big(1 - P(A_i)\big)$$

• ∏ means multiply all the terms together.
• P(Aᵢ) is the chance of event number i, with all n events independent.

In plain English: it's usually easier to work out the chance that none happen and take that from 1.

Try it

Pen and paper: a Match Odds market shows home 1.80, draw 3.90, away 5.00. What is the fair price of "not home" (draw or away), and does the book add up to more than 100%?

Answer

Implied chances are 55.56%, 25.64% and 20.00%, which add to 101.20%. Draw-or-away is 25.64% + 20.00% = 45.64%, a price of 1 ÷ 0.4564 ≈ 2.19. Taking the overround out first (divide each by 1.012) gives 45.10%, a fair price of about 2.22.

Common mistakes

• Adding chances that overlap. "55% over in one game plus 50% over in another" is not 105%. Subtract the overlap, or use 1 minus the chance of neither.
• Multiplying linked events. Home win and over 2.5 in the same match are connected. Multiplying their chances misprices the combination, sometimes by a lot.
• Forgetting the overround. Implied chances from a real market add to a bit more than 100%. Remove the margin before you treat them as fair (Lesson 2.2).
• Treating a lay as a different bet. Laying the away side is backing home-or-draw. Price both routes and take the better one.
• Thinking "at least one" is easy to get. Five independent 20% shots give only 1 − 0.8⁵ ≈ 67% of at least one landing, not 100%.

These rules are the starting point for every model on the site, and the reason most of them still lose money is rarely the arithmetic.