A Shot’s Probability Distribution

A cross court forehand's probability distribution; pretty small since it's hit by an accurate player

Tennis is random. Each shot is best understood not as a particular result (winner, error, rally ball, etc.), but as a probability distribution of possible results, from which a particular result is chosen at execution time.

When a player selects a shot, they pick a target. They aim at a particular point on the court, and they aim a certain height over the net. These choices are not always fully conscious, but players generally have an idea of the shot they’re attempting as they strike the ball.

After this decision is made, the distribution of possible outcomes is set. The image above is a visualization of that distribution.

Mapping The Territory

A probability distribution is a tool we use to describe a range of outcomes. The diagram above indicates where the ball is likely to land after the shot depicted, a cross court forehand (our pink guy is right-handed), is selected. The target, where the player is aiming, is in red, and the concentric yellow circles represent tiers of likelihood for where the ball will land. The darker the yellow, the more likely, while the lighter the yellow, the less likely.

Unless a player has a systematic inaccuracy built into their stroke (for example, they’re always late), the densest part of the distribution is at its center, close to the target. That’s where the ball is most likely to land. "Most" is relative; for an inaccurate player, it’s still not very likely to land there, it’s just more likely to land there than anywhere else. The farther from the target we travel, the less dense the distribution gets.

The exact probability density that each circle represents isn’t critical to understanding the core idea, and the numbers will vary significantly between players. That said, if numbers are helpful, assume that the inner circle is where roughly 50% of balls land, the middle ring is where an additional 30% of balls land, and the outer one is where a final 18% land (leaving an extra 2% allocated to wild mishits and whiffs). Also, the regions are cumulative – the chance that the ball lands in either the inner circle or the middle ring is 50% + 30% = 80%.

Two probability distributions for the same cross court forehand; one narrow (accurate player) and one wide (inaccurate player)
An accurate player (left) vs an inaccurate player (right) attempting the same shot. Note that the accurate player produces a much tighter distribution; there’s far less variability in where their shot will land.

Skill Level Influences Shot Selection

The breadth of a shot’s distribution varies wildly with the player’s skill. It is essential to be realistic about the distributions that our own shots are generating. If we lie to ourselves, if we convince ourselves that our distributions are smaller than they really are, we’re going to end up aiming for bad targets and missing a lot.

You’ll notice that a large amount of our inaccurate player’s distribution is out of bounds; he’s very likely to miss this forehand. In fact, our player on the right is so inaccurate that he should never be using this target; in order for enough of his distribution to be in bounds to frequently continue the rally, his just need to aim down the middle.

In contrast, this is a great target for our accurate player on the left. Both the most and second most likely region are almost entirely within the singles court, and even many of the 18% most inaccurate balls they hit will still go in.

The vast majority of players would see immediate improvements if they would humbly observe where their attempted shots really land.

First, Just Stop Missing

Missing in tennis is really bad. The player who misses loses the point immediately, with nothing at all demanded of their opponent. Further, nearly all recreational players overestimate their skill and therefore underestimate the spread of their distributions. This means that most recreational players (and many competitive ones, for that matter) routinely aim for bad targets, for which too much of the probability mass of the distribution is out of bounds.

The vast majority of players would see immediate improvements (in their singles game, specifically) if they would humbly observe where their attempted shots really land after choosing certain targets, and then adjusted those targets accordingly. Be honest with yourself, and then generate a realistic model of your shot’s possible results in your mind. Once you do that, pick targets such that 80-90% of those results are in. Try it next time you’re on the court, and you might find that, suddenly, it’s your opponent who’s doing all the missing.

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