Probability to Win a Match
In this article I will calculate the probability to win a match given an independent and identically distributed probability to win a single rally. In a previous post I already calculated the probabilities to win a game, I will extend the calculation to winning a complete match.
Independent and Identically Distributed
Rally outcomes are assumed to be independent and identically distributed. See the previous post on the probability to win a game for further discussion. We will assume that every rally will be won with a probability of \(p\) by the first player or pair. The assumption extends to games, if individual rallies are independent and identically distributed then games will also be independent and identically distributed.
Match Probability dependent on Game Probability
In the current scoring system, there are three ways to win the match:
- Win two games,
- Win one game, lose one game, win the third game,
- Lose one game, win the subsequent two games.
This leads to the formula
\[\begin{eqnarray} P_{Match}(p) &=& P_{Game}(p) \cdot P_{Game}(p) \\ &+& P_{Game}(p) \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \\ &+& (1-P_{Game}(p)) \cdot P_{Game}(p) \cdot P_{Game}(p) \end{eqnarray}\]or in shorter form
\[P_{Match}(p) = P_{Game}(p)^2 [1 + 2 (1-P_{Game}(p))]\]The first factor gives the probability to win the two needed games, the sum gives the different ways this can be achieved.
Thus the probability to win a match depending on the probability to win a game can be easily calculated.
This curve shows a rather small magnification effect. Going from a game probability of 50% to a game probability of 60% only increases the match probability to about 65%.
Fractions of different game progressions
We can also look at the different match progressions as mentioned above. The game can be won in two straight games or after losing one of the first two games and winning the other one and the third game. In the following plot, won games are denoted by W, while lost games are denoted by L.
The highest probability for a three game win is given for a game probability of around 65%. Below that the probability for a win decreases, above that the probability for a two-game victory increases. The relative probabilities in a won match are shown in the following plot.
The lower the game probability, the higher the fraction of three way victories. Approaching a game probability of 100%, all matches tend to be two-game victories as can be expected.
Match Probability dependent on Rally Probability
Combining the formulas for \(P_{Game}(p)\) from the previous post and the formula from above for \(P_{Match)(p)\) gives the probability to win a match depending on the probability to win a rally. For the sake of brevity I will not write it here. The corresponding curve for the probability of winning a match is given in the following plot.
This curve shows a greater magnification effect than the other plots. This is to be expected as the curve combines the magnification effect of the dependence of the match probability on the game probability as well as the dependence of the game probability on the rally probability.
We can also divide the won matches by their game progressions.
We see that the percentage of won three-game matches never exceeds 30%. This maximum is for a rally probability of about 55%, which corresponds to a game probability of 70%, that was found to maximize the fraction of three-game victories earlier. We can also plot the relative fractions of the different progressions of all won matches.
For small values all progressions are equally likely. In the formula above, \(1 - P_{Game}(p)\) is very close to one, so all three paths have the same probability. Above 40% the fractions of thrid-game victories declines and reaches zero around 70%. Above 70% third-game victories are negligible and all matches are won in straight games.
Table of Match Probabilities
As a reference, we can also list the probabilites to win a match for rally probabilities from 20% to 50%. Also the frequencies of won matches are listed. For example, for a rally probability of 45%, a player would win 16.1% of matches, or would on average win one match out of every 6.2 matches.
| Probability for Rally | Probability for Match | Matches per Match Won |
|---|---|---|
| 20% | 0.0000% | 9235806937.2304 |
| 21% | 0.0000% | 1834053084.2217 |
| 22% | 0.0000% | 402894587.8836 |
| 23% | 0.0000% | 97092645.5728 |
| 24% | 0.0000% | 25482478.7016 |
| 25% | 0.0000% | 7237662.8399 |
| 26% | 0.0000% | 2212239.2585 |
| 27% | 0.0001% | 724109.6283 |
| 28% | 0.0004% | 252709.9484 |
| 29% | 0.0011% | 93671.3866 |
| 30% | 0.0027% | 36750.7222 |
| 31% | 0.0066% | 15214.9938 |
| 32% | 0.0151% | 6628.9351 |
| 33% | 0.0330% | 3032.0168 |
| 34% | 0.0688% | 1452.7864 |
| 35% | 0.1374% | 727.8213 |
| 36% | 0.2627% | 380.5952 |
| 37% | 0.4821% | 207.4248 |
| 38% | 0.8499% | 117.6601 |
| 39% | 1.4413% | 69.3808 |
| 40% | 2.3539% | 42.4822 |
| 41% | 3.7061% | 26.9826 |
| 42% | 5.6306% | 17.7600 |
| 43% | 8.2628% | 12.1024 |
| 44% | 11.7232% | 8.5301 |
| 45% | 16.0969% | 6.2124 |
| 46% | 21.4121% | 4.6702 |
| 47% | 27.6240% | 3.6200 |
| 48% | 34.6058% | 2.8897 |
| 49% | 42.1531% | 2.3723 |
| 50% | 50.0000% | 2.0000 |