Bosko said:
In my opinion two parameters are needed:
p - the probability that the first (red) player wins on his serve
q - for the other player (blue) to win on his serve
Of course:
1-p and 1-q are the corresponding "loss" probabilities
Have you read the post#8 above (by erland). Outside of exception (like a tiebreak for which I don't know the full rule off-hand), in normal games you just have one player serving through-out a game (note that "game" is just part of a "set" and sets form a "complete game"). So essentially there is one parameter ##x## (which could be taken to be the probability of serving player winning a point or the probability of receiving player winning). The other quantity would just be ##1-x##.
Just in case you were still unsure here is how the game progresses. The player P1 below is serving though-out the game.
D0:
initial state is (0,0)
D1:
If P1 wins point we move to (15,0)
If P2 wins point we move to (0,15)
D2:
If P1 wins point from (15,0) we move to (30,0)
If P2 wins point from (15,0) we move to (15,15)
If P1 wins point from (0,15) we move to (15,15)
If P2 wins point from (0,15) we move to (0,30)
D3:
If P1 wins point from (30,0) we move to (40,0)
If P2 wins point from (30,0) we move to (30,15)
If P1 wins point from (15,15) we move to (30,15)
If P2 wins point from (15,15) we move to (15,30)
If P1 wins point from (0,30) we move to (15,30)
If P2 wins point from (0,30) we move to (0,40)
D4:
If P1 wins point from (40,0) then
P1 wins the game
If P2 wins point from (40,0) we move to (40,15)
If P1 wins point from (30,15) we move to (40,15)
If P2 wins point from (30,15) we move to (30,30)
If P1 wins point from (15,30) we move to (30,30)
If P2 wins point from (15,30) we move to (15,40)
If P1 wins point from (0,40) we move to (15,40)
If P2 wins point from (0,40) then
P2 wins the game
D5:
If P1 wins point from (40,15) then
P1 wins the game
If P2 wins point from (40,15) we move to (40,30)
If P1 wins point from (30,30) we move to (40,30)
If P2 wins point from (30,30) we move to (30,40)
If P1 wins point from (15,40) we move to (30,40)
If P2 wins point from (15,40) then
P2 wins the game
D6:
If P1 wins point from (40,30) then
P1 wins the game
If P2 wins point from (40,30) we move to (40,40)
If P1 wins point from (30,40) we move to (40,40)
If P2 wins point from (30,40) then
P2 wins the game
D7:
If P1 wins point from (40,40) then we move to (A,40)
If P2 wins point from (40,40) then we move to (40,A)
If P1 wins point from (A,40) then
P1 wins the game
If P2 wins point from (A,40) then we move to (40,40)
If P1 wins point from (40,A) then we move to (40,40)
If P2 wins point from (40,A) then
P2 wins the game
======================
Regarding the probability of winning from 40-40 there was this thread few years ago:
https://www.physicsforums.com/threads/winning-tennis-with-probability-0-3-and-0-7.1044416/
For this question, I did my own calculation in spreadsheet (few years ago) and remember getting the same answer as posted in question. Since I haven't re-checked it again, I thought I would look at it again and before posting the expression for probability of winning the game from start (but hopefully it isn't difficult to see how to derive it from post#9).