What are the odds of 4 aces in a row?

The top ATP server will have about a 10% probability of hitting an ACE on any first serve. So the probability of hitting four straight first serve aces is extremely small.

(.1)(.1)(.1)(.1)= 0.0001= 1 in 10,000.

Odds get better when the returner doesn’t move.

There is no reason not to assume that each first serve is an independent event and randomly distributed... Even more so if it is restricted to hitting 4 first serve aces at the start of a game.

If we assume 80 matches a year and 100 first serves per match, there will be 8000 first serves a year. Should expect it to happen about once a year -- four consecutive first serve aces in a game. At any point during the game, but second serve aces do not count.

This 2010 video claims Fed had done it six times over nine years. A bit lower than expected... And video also includes second serve aces, which we agree may not likely randomly distributed as first serve aces, because it can be argued that a second serve ace is more likely at 40-0 than at 0-40.

Probability against Draper would be much different.

Try the math(s) on these ace records:

Since 1991 (when the ATP started keeping Ace records), several guys have hit 5 aces in a row in a game --after being down 0-40. Becker did it in '95. In the past decade, Opelika did it once and Isner pulled it off twice.

For consecutive aces, Federer, Roddick (2x) and others have hit 7 aces in a row. (Obviously more than one game). Stepanek hit 8 in a row in '06. But Sam Querrey beat that the following year -- he has the record for most consecutive aces with 10 aces in a row (in '07).

For consecutive aces, Federer, Roddick (2x) and others have hit 7 aces in a row. (Obviously more than one game). Stepanek hit 8 in a row in '06. But Sam Querrey beat that the following year -- he has the record for most consecutive aces with 10 aces in a row (in '07).

Interesting. It looks like those records include second serve aces and no records are kept for consecutive first serve aces...

Would you agree that 1/10,000 is a good estimate for the probability of a top ATP server hitting four straight first-serve aces?(.1)(.1)(.1)(.1)= 0.0001= 1 in 10,000.

Of course each first-serve will be influenced by other factors such as strength of opponent, surface, weather, etc. So the event is not as clean and independent as a 50% coin flip... But over the course of a long season those factors balance out, so 1 in 10,000 estimate for a top server sounds reasonable.

Fed apparently accomplishes 4 aces in a row about twice a year. But that is including both first and second serves.

I read a profile of an NBA player who liked to practice shooting free throws. Many times he'd sink 100 in a row. Suppose his free throw average in games was 80%, it would be virtually impossible for him to sink 100 in a row if each throw had only and 80% chance of going in, (0.8^100).

This means it must be true that a person can get in a 'groove', where their chance of success is much higher than if they attempt it 'cold'.

This also means that the gamesman gets a big advantage by creating a delay when his opponent gets 'hot'. Djoker's opponent should have faked an injury after the first two aces.

Tennis Tip/Instruction: Don't be like Djoker, it's statistically too impossible.

There is no reason not to assume that each first serve is an independent event and randomly distributed... Even more so if it is restricted to hitting 4 first serve aces at the start of a game.

If we assume 80 matches a year and 100 first serves per match, there will be 8000 first serves a year. Should expect it to happen about once a year -- four consecutive first serve aces in a game. At any point during the game, but second serve aces do not count.

This 2010 video claims Fed had done it six times over nine years. A bit lower than expected... And video also includes second serve aces, which we agree may not likely randomly distributed as first serve aces, because it can be argued that a second serve ace is more likely at 40-0 than at 0-40.

Won't the first serve ace probability go down with match time due to fatigue and also due to mental effect if opponent is strong?

I have hit only one ace in my life!

I have hit only one ace in my life!

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Interesting. It looks like those records include second serve aces and no records are kept for consecutive first serve aces...

I had assumed that consecutive aces meant no 1st fault discontinuity. In that case, only the 1st in a series could be a 2nd serve. But do not know if that is really the case with the records I posted. However this five consecutive aces from Boris Becker in 1995 appears to have no such discontinuity.... No faults in the sequence.

I had assumed that consecutive aces meant no 1st fault discontinuity. In that case, only the 1st in a series could be a 2nd serve. But do not know if that is really the case with the records I posted. However this five consecutive aces from Boris Becker in 1995 appears to have no such discontinuity.... No faults in the sequence.

It appears that the "ace percentage" stat includes both first and second serves... But if you faulted on every first serve and aced every second serve, would that be 50%? Or 100%?

Not clear whether stats are kept for strictly first serve ace percentage. That stat would be more meaningful when trying to establish the probability of four straight first serve aces.. A top ATP server like Fed playing normal competition will be at about 11% (including first and second)... Although Ivo was up to 23% playing weaker competition.

Karlovic’s schedule this year and last has been roughly twice as easy as Federer’s. Weaker opponents are, on average, easier to ace. For instance, the last time Federer and Karlovic played each other, Federer’s ace percentage was higher than usual, at 11.5 percent, while Karlovic’s was lower, at 19.1 percent.​

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What are the odds with Tomic receiving with the racquets handle?

There is no reason not to assume that each first serve is an independent event and randomly distributed.

If we assume 80 matches a year and 100 first serves per match, it should happen about once a year, four first serve aces in a game.

This video includes both first and second serve aces. Apparently Fed had done it six times over nine years.

There are plenty of reasons not to assume iid. A player may be more or less willing to try to get a racket on the server after he's been aced a couple times in a row. Whatever caused a couple aces in a row to happen can also increase the odds of getting an ace on the next point. Etc. If these factors all cancel out more or less, then assuming iid can work in practice. But I'd need more than Federer to be convinced of that.

To assume the probability of 1st Serve Ace is independent may be true for the 1st ace in a series. [But it may not be, eg aces only happens in 2nd set] To determine if subsequent aces come in a series require a total different set of record keeping. Then you could calculate the likelihood of a 2nd ace following a 1st ace and so forth for a 3rd ace after a 2nd, etc. This type of likelihood statistic needs to be compared to the independent observation estimate to determine if there is some other cause and effect than simple independence. Not volunteering to do this, but theoretically, the calculation could be made for comparison.

Whatever caused a couple aces in a row to happen can also increase the odds of getting an ace on the next point

Maybe so. But it also true that a lot of these serves are simply unreturnable.

When a great server paints the line 4 straight times it is often untouchable. There is very little a returner can do about it!