Tennis and Probability


Imagine there’s a tennis match in Kyrgyzstan between two players you’ve never heard of. You don’t know their rankings, match histories, or even what they look like, but you’re tasked with predicting who will win. The only piece of information at your disposal is player1 has a p chance of winning any given point against player2. In other words, if player1 and player2 played 100 points, on average player1 would win p*100 of these points.

Given only the information above, who do you think will win? How confident should you be they will win?


I’m sure there’s a mathematical pen-and-paper way to solve this problem. I don’t know it, so we’re going to rely on computer simulations. We can understand how likely it is player1 will win by simulating some large number of matches between both players while factoring in player1’s chance of winning any given point. Our confidence in player1 is represented by the proportion of matches she wins out of the large number simulated.

Here’s an example. Suppose we know player1 has a .51 chance of winning any given point against player2. At 15-0, she has a .51 chance of winning the next point and going up 30-0. At 0-40, she has a .51 chance of getting to 15-40, but a .49 chance of losing the point and the game.

To determine our confidence player1 will win an entire match, we might simulate 1000 such matches between player1 and player2 and count how many player1 won. For instance, she might win 586 of the 1000 matches. As a result, if all we know is player1 has a .51 chance of winning any given point, we might say we are 586/1000 = 58.6% confident she will win the match.

The model we consider does this for a specific type of tennis match. All simulated matches are best-of-3 sets with a 7-point tiebreaker played at 6 games all. What’s more, the deuce is “played out,” meaning players must win by two points to win a game. I chose this format because it’s the most common type played on the ATP and WTA tours.  

We run the simulation and graph the results. 3000 matches were simulated for 50 probabilities on the interval [.4, .6]. The exact code used for all the simulations in this post is here.

Figure 1

Figure 1 plots the probability player1 has of winning any given point against the probability she wins the match. For instance, if she has a .4 chance of winning a point, it’s very unlikely (~0% chance) she will win the match. Likewise, if she has a .60 chance of winning a given point, her probability of victory is ~100%.

As expected, if player1 has a .5 chance of winning a point, she has a 50% chance of winning the match. Based on this, we might expect a .525 chance of winning a point to yield a 52.5% chance of victory. Yet, our intuition fails to grasp the benefits of a slight edge. Just increasing the probability of winning a point from .5 to .525 causes the probability of winning the match to increase from 50% to 75%! A .025 increase in win probability at the point level leads to a 25% increase at the match level. To put this in perspective, we might say the difference between a player that wins 50% of her matches and one that wins 75% of them is the latter wins 2.5 more points out of 100 than the former. To make it even more concrete, we can surmise the difference between Pete Sampras (64 career singles titles) and Carl-Uwe Steeb (3 career singles titles) is only 2.5 points per 100!

Fully explaining Pete Sampras’ dominance and Carl-Uwe Steeb’s adequacy is far beyond the scope of our model, though. At best, we can say a player with a .525 point win rate would have a match record like Sampras’. However, this does not imply Sampras actually won 52.5% of the points he played or a player who does win that proportion will be a 14-time grand-slam champion. We are only noting similarities between the statistics of idealized players considered in the model and actual ones. Resemblance only suggests both players could have other common properties.

Still, the results are striking. Player1 only needs a .55 probability of winning a given point to win ~95% of the matches she plays. Approaching .6 almost guarantees she’ll win every time. Even a .51 probability of winning a point increases the odds of winning to ~.6.

Any tennis player will tell you the margins are slim, but now you’ve seen it computationally demonstrated. Common tennis platitudes like “every point counts” and “make them work for each point” take on additional significance. The practical implication is to focus more on individual points as opposed to games or sets when playing a match. Any competitive tennis player should already do this, but now they have an additional, semi-scientific reason to do so.

Back to Kyrgyzstan. We understand player1 has a p chance of winning a given point. If p > .5, we should be confident she will win. If p > .525, we should be really confident she will win. The same holds on the other side of .5. If p < .475, we should root for an upset. If p < .45, there’s pretty much no hope.

There’s another match happening in Zanzibar. We still know player1 has p chance of winning any given point, but now we know something else: she chokes. Whenever she has game, set, or match point, her probability of winning said point drops from p to pc. As an example, imagine her p = .55 and pc = .35. Suppose she’s also up 40-15 in a game against player2. Although her probability of winning a given point is .55, because it’s a game point, there’s only a .35 chance she wins the next point to secure the game. If she was up 6-4 in a tiebreaker, the same decrease in probability occurs. There’s only a .35 chance she wins the tiebreaker 7-4.

A similar problem presents itself. Given player1’s p and pc, will she win the match? How confident should we be in our prediction?

We run the same model in the previous section with some minor modifications. Whenever player1 is poised to win a game, set, or match, her probability of winning the next point plummets to some pc. As in the previous section, we vary p within [.4, .6] and observe player1’s probability of winning the match. What’s different now is each curve is associated with a constant pc.

Figure 2

The blue “standard” curve represents how player1 would perform sans choking. It is the same curve depicted in Figure 1 where her probability of winning a given point is p regardless of the score. The others record what a player’s match record at a certain p would be like given her pc. If player1’s pc = .45 for instance, her performance suffers heavily. At p = .5, she has only ~40% chance of victory compared to the 50% chance observed with an identical p in the absence of choking.

Past p = .5, a .05 decrease in pc roughly corresponds to a 10% reduction in the chance of winning a match. If player1’s p = .525, and pc = .45, her chances of winning would be 62%. However, if her mental game falters and pc drops to .4, her chance of victory decreases to 53%.

Clearly, any level of choking impedes performance. If we know player’s pc is low, we should expect a rather large p to compensate. Player1’s coach might suggest addressing the root factors of unclutchness. A sports psychologist or deep reflection might increase pc, or eliminate choking altogether.

There’s a third match in Andorra and we’ve learned more about player1. She no longer chokes at this junction in her career, but her play has become streaky. Given she won the previous point in a match, there’s a probability ps she will win the next point as well. If ps = .8, for example, there’s an 80% chance she will win the next point after winning the previous one. However, streakiness goes both ways. Given she lost the previous point, there’s an 80% chance she will lose the next one as well. The first point of every game (and tiebreak) is a fresh start, though. The chance she wins that point is p. From then on, ps reigns.

To get a sense of our credence in player1’s performance, we run another simulation.

Figure 3

This is a heatmap. The color of the block occupying the (probability of winning first point, ps) coordinate corresponds to the probability of winning the match. Light colors indicate high probabilities while darker colors represent low ones. As an example, if player1 has a .42 probability of winning the first point of any given game or tiebreaker and her ps = .9, her probability of winning the match is around .3. Here, we note heatmaps favor concise representation over numerical precision. The bar on the right gives us only a general idea of what probabilities are associated with which colors.

Still, patterns are clear. Variation in ps only has noticeable consequences at the extremes. If p is low, then a high streakiness almost guarantees losing the match. Observe how the probability of victory is only ~20% when p = .4 and ps > .9. However, for any value of ps < .9, the probability of winning the match hovers around 40% if p remains fixed. Similar results occur for higher p’s. Only when ps is large do we see its influence on the chance of victory. At values of p close to .5, ps has no apparent effect. The probability of winning the match stays roughly constant as ps varies.

It might look like high ps, or streakiness, is an advantage when p is sufficiently greater than .5. This is true, but the gain is comparatively small. Recall the Kyrgyzstan model where player1’s probability of winning the next point was constant throughout the match. If p = .525, player1 had 75% chance of victory. Here, for all levels of ps, p = .525’s probability of victory hovers around 60%. Player1 is much better off winning 52.5% of points in general as opposed to embracing streakiness.

We should note streakiness as we’ve defined it “smooths” performance at low values of ps. If ps = .5, this means players have a 50% chance of winning any given point after the first. In other words, we’re saying the players are just about even after the initial point. If p = .4 in the Kyrgyzstan model, a player has almost no chance of winning. In this model, p = .4 gives an ~40% chance. What it captures at these kinds of values is the dynamics of one player frequently winning the first point, and then both players having an equiprobable chance of winning the following ones.

We take these results to Andorra. Streakiness only concerns us at the extremes. If player1 has an exceptional chance of winning the first point and is very streaky, we’re confident. If she’s dismal on first points but just as streaky, we lose faith.


I’m excited about these models for two reasons. First, they suggest other interesting questions to tackle. How do the probabilities change in 5-set matches? What about no-ad? What if a player is brilliant on break points? What happens if we incorporate a serving advantage? Can we combine several of these models? The list goes on.

The second reason is epistemological. Can models of this type provide ever provide an explanation of real-world phenomenon? I talked about the Kyrgyzstan model being insufficient to explain Pete Sampras’ dominance, yet, there are distinct reasons why people may think this is so. We could say to explain Pete Sampras’ skill, we must address specific aspects of his game. It’s necessary to observe his deft touch at net, booming serve, and flat forehand. Pete Sampras is good, the reasoning goes, because he was able to hit aces and put away volleys. Any explanation of Sampras’ skill has to begin with these factors. Under this perspective, the Kyrgyzstan model fails to give an explanation because it is too abstract. Only tracking the probability of winning any given point obscures the unique advantages Sampras had that contributed to such a high probability. Proponents of this critique believe, in principle, no idealized model can explain why a given tennis player was so successful. Such models are incapable of capturing the unique, individual aspects of a player that contributed to his or her dominance.

A second critique finds no fault with idealized models in general — just the Kyrgyzstan one. It claims idealized models can give explanations of real-world phenomena, but this one in particular is too weak to do so. A model that actually provides an explanation would take parameters like probability of winning break points, probability of winning a point on second serves, probability of winning games/sets when down or up, etc… It would be much more detailed, but it still wouldn’t directly address the unique aspects of Sampras’ tennis the prior camp requires. Explanations from the model would sound like “Sampras was great because he had a high win percentage on second-serve points,” or “Sampras was a champion because he had a high probability of saving break points.” This camp believes idealized models can supply these explanations and they’re sufficient for deep knowledge. To understand Sampras, we only need to understand his propensity to win or lose in general scenarios. We only care about his serve or volleys or movement insofar as they contribute to a high break-point save rate, for instance. The statistical measures provide the base of our understanding; everything else is secondary.

We might be partial to a certain type of explanation. Players and coaches will tend to explain matches in terms of backhands and serves where statisticians will invoke probabilities and win rates. Our intuitions are often with the players. We find it hard to believe understanding Djokovic is possible without seeing a sliding open-stance backhand winner, or a kick-serve that bounces over your head in the case of John Isner.

Are our intuitions correct? Are the modelers correct? Which level of explanation gives us the best understanding of a tennis player? Is one of these levels subordinate to the other?

Consider the ideal

Imagine a young person trying to decide what to do with her professional life. Excluding financial considerations, she might consider three principles [1].

  • The principle of approbation
    • Do the thing that earns the most social praise.
  • The principle of altruism
    • Do the thing that increases the well-being of humankind.
  • The ideal principle
    • Imagine the world is perfect and your opportunities are endless. No wars, no famine, no disease, and no obstacles preventing you from pursuing whatever you wish. Do the thing you would choose to do in this alternate world.

If our young adult is independent-minded, she will dismiss or heavily discount the principle of approbation. There are two main reasons why it might get priority over others, but neither is convincing. For one, she could think picking a plan that satisfies the principle of approbation is a good bargain. Following the principle, she could only select professions that win wide social approval (doctor/lawyer/engineer). In exchange, she gets a lifetime status boost.

The discussion could end here. Different people require different levels of praise to feel content and secure. If our young adult is the type of person that needs affirmation, yielding to the principle of approbation might be a good decision. Yet, we will continue with our assumption she is an independent thinker. Her sense of self-worth is not heavily tied to approval so she doesn’t find the trade compelling. It diminishes her freedom to choose, and, in return, she receives a good she has no dire need for.

We can also think of the principle of approbation derivatively. Our young adult might actually want to help others, so she gives precedence to the principle of altruism. However, she realizes she is in a poor position to determine the best method to go about it. She might reason that others know a lot more about helping people than she does, so listening to popular opinion might be productive. Social approval could serve as a proxy for how effective her altruism is. After all, doctors help people, and people really love doctors. Praise for professional choices is actually a sign she’s on the right path.

While tempting, she does not buy this line of thought. She acknowledges the limits to her finding the best way to help others, but she has reason to believe the crowd can’t do much better. Doctors have always been popular, but their treatments weren’t always effective. Obviously, doctors now have a much better understanding of the human body, but the point is that praise and occupational prestige may not correlate well with how effective one is at aiding others.

The principle of approbation has been dismissed. Choosing between the remaining principles might look easy. Why not choose the principle of altruism and leave the world better than you found it? It’s inexcusable, the thought goes, to yield to self-indulgence when you can heal the sick and feed the hungry. Those in favor of altruism will say we face moral obligations in our professional lives. Not only must we “do no harm,” as Hippocrates might say, but we must also alleviate the suffering around us. By being of sound mind, able-bodied, and of comfortable means, we are automatically tasked with using our individual talents to help others flourish. From this perspective, the ideal principle is irrelevant. As long as people suffer, we must rush to their aid. There is no need to even think about how you would live in a perfect world, as the world is, in fact, deeply flawed.

Despite this, I think the ideal principle is crucial. We should all have an answer to “what would you do all day in a perfect world?” To be clear, I also believe the principle of altruism should weigh heavily in informing our professional lives, but neglecting where our own preferences and inclinations might lead us in an ideal scenario is a mistake. We suppress our individuality when the question goes unasked. In the same way someone who adopts the principle of approbation is subject to external forces when deciding where to put her professional energies, the principle of altruism ties our professional lives to processes beyond our control.

Note I am well aware what we can and cannot do is generally determined by things beyond our control. It is also true the claims those suffering have on us are more legitimate than those of popular opinion. These two facts make a lack of control normal and even a consequence of being moral. My only point is the principle of altruism subjects you to additional, exogenous constraints that diminish autonomy [2].

Why should we care about autonomy? The answer is that the people who we really are, the alternate selves that are the most “us,” are the people we would be in a state of absolute freedom. Imagine tomorrow you received the ability to do whatever you want. All restrictions — financial, emotional, cultural, and otherwise — are lifted. The world is perfect (it doesn’t need your help) so you can, in good conscience, choose to do what you wish. This is absolute freedom [3].

In the real world, some decisions are made for us. An authority figure could make you choose from a small selection of alternatives, or natural circumstances might do the same. The problem is we can’t infer much about an individual based on these choices. Can we really say someone is kind if they are forced to be? Is it appropriate to call someone cruel if they had to choose between three cruel alternatives? Our intuition says no [4]. It’s easier to call someone brave and courageous if they could have been otherwise. The choices we make under favorable circumstances reflect who we really are, and the ones we might make in a state of absolute freedom do so the most [5].

When we dismiss even considering the ideal principle, it’s a failure of self-knowledge. As long as we don’t have an answer to “what would you do all day in a perfect world,” we fall short of understanding ourselves. We might grasp how the world influences us, either through approbation or altruism, but we are distinct from it and deserve independent consideration. Thinking about how we would behave in absolute freedom illuminates who we are, without the external demands and obligations layered upon us.

Lacking self-knowledge poses a deep difficulty. How can we shape our lives when it’s unclear who lives them? To me, it’s like building a house without knowing the inhabitant or writing a love letter to an unknown recipient. Even if we happen to build or write for the most extraordinary people, what we create will be uninspired. We can only address their most general features (i.e. “put a kitchen in the house” or “you have a beautiful smile”) without touching anything specific to them. Building something wonderful requires knowing who it will serve. In a certain (obvious?) way, a life first and foremost serves the person who lives it.

Thinking about ourselves is natural, but it’s unnatural to embrace it. Yet, this does not mean the ideal principle should be the only criterion that guides our lives. It is a thought experiment, necessary to consider and grapple with, but not something that should (or even can) dictate your entire life. For instance, some of our ideal lives might be out of reach (playing center for the LA Lakers) while others could be mundane and a little abhorrent (watching TV all day?). In either case, we learn something about ourselves by considering the ideal principle. It could prompt conviction (“I really like sports. Perhaps I should work in the Lakers front office”) or reassessment (“Am I really that type of person? Something needs to change”).

This does not settle the matter as to how a young person should decide where to put her professional energies. I believe the ideal principle is important, but primarily as an exercise in hypothetical reasoning whose results are used to inform a broader decision. Exactly how much weight the ideal principle deserves to guide real professional conduct is up for debate. At the very least, I think two things are clear. First, the principle of approbation should be ignored in the vast majority of cases. Second, we all need an answer to the question posed by the ideal principle. After all, it might be who you are.




[1] This is far from an exhaustive list of principles we could use to make professional decisions. At varying levels of specificity, potential principles include: the excellence principle (“do the thing you’re best at”), the aesthetic principle (“do the thing that brings the most beauty into the world”), the comfort principle (“do the thing that gives you the highest standard of living”), a smattering of ideological principles (“do the thing that furthers communism/socialism/capitalism/anarchism etc…”), the contrarian principle (“do the thing people will dislike”), the hedonistic principle (“do the thing that gives you the most sensual pleasure”), and variants of a religious principle (“do the thing God/Allah/Yahweh wants you to do”).

There are a lot of principles people might deem relevant. The three I chose are among the most general, and, I’ve observed, often crop up in actual conversations young people have about their futures (barring financial considerations)

[2] But if you choose the constraints, are they even constraints at all? I’m the one who brought them about and limited my own behavior, so aren’t they best understood as artifacts of my freedom, rather than obstacles to it?

[3] Absolute freedom is a state without hardship. If you’re concerned about the effects of hardship and struggle on personal identity, remember we enter absolute freedom tomorrow. All of our past experiences, the things that have made us “us” are preserved. In fact, I don’t think a person born and raised in absolute freedom would be much of a person at all (or at least not a very good one). Considering absolute freedom is meant to make us understand how we would behave if circumstances changed dramatically, not who we would be if our pasts were different.

[4] It is true we can observe how people react to external circumstances and make inferences that way, though. It’s the mark of a strong and magnanimous soul not to harbor spite when wronged, for instance. I’m on the fence as to whether we can really understand who a person truly “is” solely by looking at these reactions. Our ability to harbor discomfort is only a part of who we are (though a very important one) and it has reasonable limits. I’m not sure what we can infer about character from someone irascible who suffers from chronic pain.

[5] There’s a “moral luck” problem here. If we think of absolute freedom as a kind of utopia, then there’s no opportunity for courageous or kind actions, and perhaps, no possibility for courageous or kind people. A solution would be to think of absolute freedom as exactly like the regular world sans obligations. War and famine still exist but somehow our obligations to stop them disappear. Now it’s possible to be compassionate in a state of freedom by deciding to be a peacemaker or go on humanitarian missions, for example.

It gets tricky when we realize what motivates this person must be completely endogenous. Their rationale for their actions won’t appeal to the principle of altruism, because that is an external obligation that cannot exist in a state of absolute freedom. Yet, if we want to call their actions kind, they must accord with some desire for the well-being of others. It looks like we’re forced to posit an “internal” principle of altruism that operates independently of the one that creates external obligations. The strength of this internal force is the measure of how kind and compassionate someone is. Yet, how is this different from the external principle? Is the internal one best understood as a heightened sensitivity to the external principle of altruism?

The other solution is to bite the bullet and say no kind actions are possible in a state of absolute freedom, as these must be motivated by the (external) principle of altruism which doesn’t exist. This, taken in conjunction with my claims in the main body, produce strange implications. Does this mean the person who is most “you” is entirely selfish? Does altruism diminish our individuality? Broader questions about utopias are also at stake. Is being an altruist in a utopia redundant? If everyone is doing fantastic, why would you care about someone else’s well-being? Your caring won’t make them do any better (the world is perfect!).