Yesterday, we talked about expected probability of winning a set between two players

Recursive Probability is even more elegant. Recursive is when a variable refers back to itself, and in this case, a parameter in this problem simply refers back to itself, which is why we call it recursive

Normally when we solve for y (such as y = 2x + 4), we generally do not expect x to be on both sides, but when a problem is recursive, x finds itself on both sides

So, what do I mean by that ?

You can think of winning a point as being in a - state. Carlos Alcaraz plays Novak Djokovic in the championship final at Wimbledon today. Imagine Alcaraz is at deuce, and so, he needs to win two more points consecutively to win the game

More importantly, winning the next point and then losing the subsequent point, or losing the next point and then winning the subsequent point - is not going to help him win the game. Both of these outcomes will simply return the score to deuce ~ which is our initial state (the state we started from)

Awesome

**So, how would we think about the probability of Alcaraz winning a game, when he stands at deuce, on his serve ?**

Alcaraz wins about 75% of his first serve points and 54% of his second serve points on grass (ATP Tour stats here), let us assume he is just as likely to make a first serve as he is to miss it, and when he misses it, he is going to need a second serve

**Probability of Alcaraz winning a point when he is serving on grass :**

= (1/2 * 75/100) + (1/2 * 54/100)

= 64.5/100

= ~ 6.5/10

I wish I had Alcaraz’s stats against Djokovic alone, but I do not. I only have their stats when they met in the championship final last year, but that is too small a sample size for me to be confident about it. So we’ll go with what we have

So, when Alcaraz is on serve, he wins 6.5/10 points. So, that means he loses 1 - (6.5/10) = 3.5/10 of total points played when he is serving. **This is the foundation from which we would build this problem **

Great, let us define our p

**p in this case is the probability of Alcaraz winning the game when he is at deuce **

**Three things can happen, right ? **

**Alcaraz can win both points consecutively and win the game, and the probability of that is p x p**

= (6.5/10) x (6.5/10)

= 42/100

**Alcaraz can win the first point and then lose the next one, and the probability of that is p x (1 - p)**

= (6.5/10) x (3.5/10)

= 23/100

and here is the sheer mathematical elegance of it - when this happens, he is back at deuce, and hence this 23/100 essentially becomes (23/00) x p

Why ? because we defined p as the probability of Alcaraz winning the game when he is at deuce (and now, he is at deuce). When Alcaraz lost the second point, he simply came back to his initial state, and parameter p refers back to itself (and hence we call it recursive)

Hence, we write (23/100) as (23/100) x p

**Third and finally, Alcaraz can also lose the first point and then win the next one, and the probability of that is (1 - p) x p**

= (3.5/10) x (6.5/10)

= 23/100

= and this also becomes, (23/100) x p

**Now what ? **

**Well, let us solve it **

p = probability of (outcome 1 + outcome 2 + outcome 3)

p = (42/100) + (23/100)p + (23/100)p

**Notice there is p on both sides, it refers to itself and that is the recursive nature of this problem. It is draped in mathematical elegance**

p = (42/100) + (46/100)p

p - (46/100)p = 42/100

(54/100)p = 42/100

p = 42/54

**p = ~ 7.7/10**

So, given what we know about Alcaraz’s stat of winning 6.5/10 points on serve, there is a ~ 7.7/10 likelihood that he is going to win the game, when he stands at deuce

**We can use the same probabilistic framework to solve other problems too **

We can easily apply this framework to study historical stats too : one of the best ever to hold a tennis racket in the All England Lawn Tennis Club is Pete Sampras, and he won 85% of first serve points on grass (stat here). If Sampras is at deuce and he decides to get two first serves in, there is a 9.7/10 likelihood that he would win the game

As we get stats that pertain to only Djokovic v Alcaraz, and on grass (and preferably for only their matches in SW19), the probabilities we estimate will tend to the true value more and more with time

Enjoy the finals !