Tennis Point Probability

Information available:

Player A wins 70% of his service points, and 40% of his return points
Player B wins 72% of his service points, and 37% of his return points.
Population average win on serve: 65%
Population average win on return: 35%

Given this information is it possible to deduce the liklihood Player A wins on serve vs. Player B, and vis versa?

For Player A serve, I was trying 35% / 37% * 70% = 66.2%

to test, I tried Player B return: 65% / 70% * 37% = 34.4%

and since 66.2% + 34.4% <> 100%, I'm obviously missing something. Any help?! Thanks in advance.

I don't see how you got that. In order for A to win he can:
1) Win on his serve. Probability .70..
2) Win on his first return. So he would have to NOT win on his serve, Probability 1- .7= .3, B not win on his return, Probability 1- .37= .63, A win on his return: Probability .4. Probability of winning on first return: .3(.63)(.4).
3) Win on his second serve. Not win on his serve, Probability .3, B not win on his return, Probability .63, A NOT win his first return, Probabilty .4, B not win on second return: Probability .63, A win on second return: Probability .4: probability A wins on second return, .3(.63)(.4)(.63)
4) Win on his third serve. .3(.63)(.4)(.63)(.4)(.63)

It should be clear, now, that "win on his nth serve" is .3(.63)= .189 multiplied by (.4)(.63)= .256 n times.

So the probability of A winning on his serve or any return is \(\displaystyle .189+ (.189)(.256)+ (.189)(.256)^2+ (.189)(.256)^3+\cdot\cdot\cdot = .189\sum_{n=0}^{\infty} .256^n\). That is a "geometric series" and there is formula for such a sum: \(\displaystyle \frac{.189}{1- .256}\).

I don't think you understand my question correctly or I'm not understanding you, thanks nonetheless. Player A wins 70% of his serves against ALL opponents, Player B is clearly an above average returner, so I would expect Player A to serve below 70%. But how do you calculate that given the information that I have?

If you have no idea how well A plays against B, there is no way you can do this.

So if they've each played the same opponents and not each other you cant come up with a probability of winning a serve against each other? Seems like a ridiculous assumption.