Conditional probability

The question

"The home side has just scored. The price went from 2.15 to 1.40. Is that the right move, or an overreaction?"

How does a goal change the odds? A goal changes the question the market is answering. Before kick-off it's "who wins from 0-0 over 90 minutes?" After the goal it's "who wins from 1-0 with an hour to play?" Conditional probability is the tool for answering the second question properly.

The idea in one sentence

Once something has happened, you throw away every future in which it didn't, and share 100% out among the futures that are left.

The picture

Picture 1,000 copies of the same match played out from the kick-off. Each copy ends differently. Before the match, the home side wins in about 464 of them.

Now the home side scores on 30 minutes. Most of those 1,000 futures had no home goal on 30 minutes, so they're gone. You keep only the copies that match what you've seen, and ask how often the home side wins in those.

That's all "given" means. The chance of a home win given 1-0 on 30 minutes is a new question with a new answer, and it's much higher.

Moment	Home win	Draw	Away win	Over 2.5
Kick-off, 0-0	46.4% (2.15)	25.8% (3.88)	27.8% (3.60)	48.2%
1-0 on 30 minutes	74.0% (1.35)	18.3% (5.46)	7.7% (12.99)	51.7%
0-0 on 30 minutes	40.8%	33.2%	26.0%	25.2%

Model chances with fair odds in brackets, from a simple goals model with home expected goals 1.5 and away 1.1. Illustrative, not a live match.

Look at the bottom row too. Nothing happening is also information. Thirty goalless minutes pushes the draw up and cuts over 2.5 roughly in half, which is why an under 2.5 back gains value with every quiet minute.

Worked Betfair example

The model: home expected goals 1.5, away 1.1 over 90 minutes. Goals arrive at a steady rate, so each team's expected goals for the rest of the match is its full-match figure times the share of time left. (Illustrative inputs, and a simplification: real matches have injury time and slightly more goals late on.)

Part 1: the goal on 30 minutes

1. Time left. 60 of 90 minutes, a share of 60 ÷ 90 = 0.667.
2. Expected goals from here. Home 1.5 × 0.667 = 1.00. Away 1.1 × 0.667 = 0.73.
3. What a home win now needs. The home side is already one up. They win if they score at least as many as the away side from here on. It's a draw if the away side scores exactly one more.
4. Add up those futures. Using the Poisson chances for each team's remaining goals, the home side wins 74.0% of the futures left, a draw is 18.3% and an away win 7.7%.
5. Fair prices. Home 1 ÷ 0.740 = 1.35, draw 5.46, away 12.99. Before the goal they were 2.15, 3.88 and 3.60.
6. Compare with Betfair. The home side is trading at 1.40. Expected value of £50 at 1.40 after 2% commission: 0.740 × £20 × 0.98 − 0.260 × £50 = £14.50 − £13.02 = +£1.48, about +3p per £1.
7. Break-even price. With 2% commission you need at least 1.36 to break even at a 74.0% chance.

Verdict: by this model the 1.40 is slightly generous. The honest reading is that the gap is small, and a simple model that ignores how a leading side plays is more likely to be wrong than the whole market is. Treat a small gap like this as a question, not an order.

Part 2: goalless at half-time

Over/Under and First/Second Half Goals are where conditional probability is most visible. Take a match expected to produce 2.6 goals, split evenly at 1.3 per half.

1. Pre-match over 2.5. With 2.6 expected goals, the chance of three or more is 48.2%, fair odds 2.08.
2. Chance of 0-0 at half-time. e^(−1.3) = 27.3%.
3. Chance of 0-0 at half-time AND over 2.5. This needs three or more in the second half: 27.3% × 14.3% = 3.9%.
4. Divide. P(over 2.5 given 0-0 at half-time) = 3.9% ÷ 27.3% = 14.3%. Fair odds jump from 2.08 to 7.00.

Verdict: the over 2.5 price should roughly treble at a goalless half-time. If you backed it pre-match, your bet hasn't become unlucky. It's become a different, worse bet, and the price says so.

The formula

The definition

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

• P(A | B) is the chance of A given that B has happened. The bar reads "given".
• P(A and B) is the chance both happen.
• P(B) is the chance of the thing you've seen.

In plain English: out of all the futures where B happens, what share also have A? Divide by P(B) to scale what's left back up to 100%.

The multiplication rule, the right way round

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

In plain English: the "and" rule from Lesson 3.1 always works if you use the conditional chance for the second event. Plain multiplication is the special case where P(A | B) = P(A), which is what independent means.

Remaining goals in play

$$\lambda_{\text{left}} = \lambda_{90} \times \frac{90 - t}{90}$$

• λ₉₀ (lambda) is a team's expected goals over the full match.
• t is the minute now.

In plain English: at a steady scoring rate, a third of the match left means a third of the goals left. Real models tilt this for the late-goal effect and injury time.

The in-play home win

$$P(\text{home wins} \mid h\text{-}a) = \sum_{i,j} P(i)\,Q(j) \,\big[\,h + i > a + j\,\big]$$

• h and a are the current home and away goals.
• P(i) and Q(j) are the Poisson chances of the home side scoring i more and the away side j more.
• The square brackets equal 1 when the home side finishes ahead, and 0 when not.

In plain English: list every way the rest of the match could go, and add up the ones where the home side ends in front. Lesson 4.2 builds the full grid.

Try it

Pen and paper: the same match (1.5 v 1.1) is 0-0 on 60 minutes. What are the expected goals left for each side, and what's the chance nobody scores again?

Answer

A third of the match is left, so home 1.5 ÷ 3 = 0.50 and away 1.1 ÷ 3 = 0.367, a total of 0.867. The chance of no more goals is e^(−0.867) ≈ 42.0%, which is also the chance it ends 0-0.

Common mistakes

• Mixing up P(A | B) and P(B | A). The chance a team wins given they scored first is not the chance they scored first given they won. Swap them and you'll misprice everything (Lesson 3.3).
• Thinking quiet minutes change nothing. No goal is information. Every goalless minute moves the under, the draw and the Correct Score 0-0 in their favour.
• Treating the scoring rate as fixed after a goal. Leading teams often sit deeper and trailing teams push. A steady-rate model is a starting point, not the answer.
• Calling a price move an overreaction without doing the sum. A home price going from 2.15 to 1.40 after a 30th-minute goal looks huge. The maths says roughly that move is right.
• Forgetting injury time. A "90-minute" match runs longer, and the second half usually has more goals than the first. Build that in before you trust small edges.

The biggest in-play errors aren't in the arithmetic but in the inputs, which is the same trap covered in why good models still lose money.