I was wondering how to combine some odds, and I was hoping someone here could help me. This does apply to poker, but my problem is easiest to explain through another sport.

Let's say two people are playing tennis. Player A wins in straight sets 10% of his wins, 90% of the time he wins in a different way. Player B loses in straight sets 80% of the time, and 20% of the time he loses in some other way.

How do I combine these odds to find out the odds of when they play Player A winning in straight sets?

but I'd like to know the mathematics the synthesizer goes through. Save myself 5 bucks too.

Thanks for the response.

Not enough info.

What info would you need?

Set wins are a derivative of the match odds, get them, and from there its very simple.

Say you have a match with two players who are even in ability and are both priced by a bookmaker at 5/6, representing 'pure' odds of 1/1, with a margin added for the bookmaker to profit.

Their set victories prices are a derivative of their match odds, with the shorter price of the result generally being with whoever wins the first set. In a three set match between these two players, a typical set break down would look like this:

Outright: Player A - 5/6, Player B - 5/6.

Player A to win 2-0: 9/4
Player A to win 2-1: 11/4
Player B to win 2-0: 9/4
Player B to win 2-1: 11/4

Note that the percentage chance of 9/4 and 11/4 adds up to approx the same chance as 5/6 - roughly 55%. Again, the 'pure' odds in this case would add up to 50% if the bookmaker were to take no margin, most likely something like 5/2 for 1-0 victories and 7/2 for 2-1.

The stats you have mentioned are completely useless from a pricing point of view without more information about the players relative abilities, as any effect the % of loss rate in certain scenarios is totally dominated by the skill level of one over the other.

For example, you could have an amateur who wins 10% of their games in straight sets, and a journey man pro who loses 80% of their games in straight sets, but if you put them against each other, the amateur will probably not even win a single game let alone set regardless of their track record.

This may be helpful:

http://web.williams.edu/go/math/sjmi...Loss_Paper.pdf

However it's only an estimate. There's no straightforward way to just take two players' winning percentages against a general field and apply it to them playing each other. The reason is that an e.g. 60% winrate could mean at one extreme that I always beat the bottom 60% of players and always lose to the top 40% (step distribution), or at the other extreme that I beat every player with a probability of 60% (uniform distribution). Or something in between.

So you need to know more than just the winrate against the overall field if you want to have an accurate answer.

To add on to DarkMagus' point, individual matchups could change things completely. Team A could be a 90% winner and Team B a 10% winner against teams in their league. However, Team B is a 95% winner against Team A because in some unique way they matchup well against Team A. It would probably be hard to find such extremes in practice for most sports but in theory such situations (particularly less extreme ones) can exist.