Predicting the Premier League

For each of these three matches, there are 3 possible outcomes: Win, Draw, or Lose. Assuming the results are completely independent from each other (e.g., Al Ahly winning doesn't force Zamalek to lose), we have $3 \times 3 \times 3 = 27$ different possible endings.

That leaves only 2 scenarios out of the 27. Which means we can divide 2 by 27 directly to get $\approx 7.4\%$.

This is a simple application of combinatorial probability. The probability of an event equals the number of desired outcomes divided by the total number of possible outcomes.

But what is the problem with this method? We implicitly assumed the probability of a Win, Draw, and Loss are all equally likely for all 3 teams ($33\%$). Is the probability of Al Ahly winning exactly the same as them losing? The answer is mostly no. That's why we need to look at the probabilities of events individually.

If we assume, for example, that Al Ahly's win probability is $70\%$, a draw is $20\%$, and a loss is $10\%$, the overall probability of winning the league is no longer $2 \div 27$. Each scenario now has a vastly different weight. We need to multiply the actual probabilities of the specific required outcomes.

But where do we get these probabilities? From guessing? No, we use available facts and data: the team's form over the last 5 matches, win ratios, historical direct matchups, and average goals scored and conceded.

We've extracted the variables, calculated the probabilities, and arrived at a final result of $6.1\%$. Beautiful. But what if these numbers changed, whether slightly or significantly? What if we tried all possible combinations thousands of times to see what happens?

Initially, the model threw a curveball: Al Ahly had a $0\%$ chance, but there was a massive $23\%$ chance of a "Tie". The model didn't understand the league's head-to-head tiebreaker rules! It treated tied points as an "unknown result". Once we fed the tiebreaker rules into the engine, the unresolved ties collapsed into the final distribution, perfectly matching our mathematical calculation.

Up until now, we've treated the probabilities as static. As if the world doesn't change. But what if new information appears before the match? A sudden injury? A tactical change? Do we stick to the old numbers, or do we update our conviction entirely?

This brings us to the most beautiful philosophical part of our analysis: What is the probability of Al Ahly winning this match, GIVEN that they won the league? Because winning this match is a strict requirement for winning the league, this probability (the Likelihood) is exactly $100\%$ ($1.0$).

The equation becomes brilliantly simple: $1 \times 0.062 / 0.441 \approx 14\%$. The probability almost doubled. Here lies the secret beauty of Bayes' Theorem; based on a single piece of evidence, you shift your entire paradigm.