Advanced probability formula help for a relative layman

[AJW82]

Hi,

I have my GSCE in maths but since then have not been that involved with it. Recently I've got very interested in probabilities, statistics with particular reference to modelling tennis match outcomes. I'm struggling to break down and understand the below and what I'd like to do is understand it so that I can place it in excel.

I’m looking at a model where we have two player’s, A and B, and player A has a constant probability p of winning a point. With some following instructions from the paper I'm referring to.

"We set up a Markov chain model of a game where the state of the game is the current game score in points (thus 40-30 is 3-2). With probability p the state changes from a, b to a + 1, b and with probability 1 − p it changes from a, b to a, b + 1. Thus if P(a, b) is the probability that player A wins when the score is (a,b), we have:

P(a, b) = pP(a + 1, b) + (1 − p)P(a, b + 1)"

(In this example let’s assume the score is 0-0, I.e. 0,0 and p = 0.74 (74%))

Also it satiates the following - the boundary values are P(a, b) = 1 if a = 4, b ≤ 2, P(a, b) = 0 if b = 4, a ≤ 2 – and not too sure what that means.

In large part the reason I’m struggling to understand how the formula breaks down as I’m not clear on what the commas mean etc. Any help or pointers would be greatly appreciated and I hope I've posted in the right section.

Thanks,
Alex

 

[mmwave]

Hi Alex,

It's great to hear that you have developed an interest in probabilities and statistics, especially in the context of modeling tennis match outcomes. Breaking down and understanding the formula you have provided is definitely a key step in being able to apply it in Excel.

First, let's define some key terms in the formula:

- P(a, b): This represents the probability of player A winning when the score is (a, b). So for example, P(3, 2) would represent the probability of player A winning when the score is 40-30.
- p: This represents the constant probability of player A winning a point. In your example, p = 0.74 (74%).
- a: This represents the number of points player A has won in the current game.
- b: This represents the number of points player B has won in the current game.

Now, let's break down the formula itself:

P(a, b) = pP(a + 1, b) + (1 − p)P(a, b + 1)

This formula is saying that the probability of player A winning when the score is (a, b) is equal to the probability of player A winning when the score is (a + 1, b) multiplied by the constant probability p, plus the probability of player A winning when the score is (a, b + 1) multiplied by the probability of player B winning (1-p). This is essentially saying that in order for player A to win the game, they must win the next point (a+1, b) with probability p, or player B must lose the next point (a, b+1) with probability (1-p).

Now, let's look at the boundary values:

- P(a, b) = 1 if a = 4, b ≤ 2: This means that if player A has won 4 points (score of 40) and player B has won 2 points (score of 30 or less), the probability of player A winning the game is 1 (100%). This makes sense, as winning 4 points in a game means you have won the game.
- P(a, b) = 0 if b = 4, a ≤ 2: This means that if player B has won 4 points (score of 40) and player A has won 2 points (score of 30 or less