There is good news and bad news. The good news is I am playing tennis regularly (as regular as one can expect from an unstructured person like me). The bad news is I get whipped by my coach every day – I practice and during warm up it looks like I win one of three points while he wins two points. But, then when we play a set, and I lose 6-1 or 6-0 most of the time.

After an energetic 6-0 drubbing again today, the exercise had its desired effects. I was sweating, my heart muscles were stronger, and my mind was whirring like a Rolls Royce engine. I then started wondering about why I lose 6-0 or 6-1 fairly regularly. No, there are no nerves or lack of mental toughness – if anything, being the underdog, I consistently play as well as I play in practice and would hazard to say I am still winning one of three points! My first uninformed reaction was I should be winning one of three games so I should have a score of 6-3 more often than not.

After thinking about this for some more time and then doing some math in my head for ten odd minutes, it was clear that my odds of winning a game were less than 1/5 (closer to 1/7 based on exact calculations). So let me explain why this is the case.

I am going to simplify the game of tennis for the purposes of this probability calculation. Let us pretend that a game of tennis is won when a player has two more points than the opponent – in other words let us pretend that we have some how reached deuce and find out the probability that I win a game from deuce. Recall that Advantage to me, means I am one point ahead, and Game is when I am two points ahead. If my opponent wins the point when it is advantage to me we go back to deuce. The following video shows Andy Murray (who has promised a lot but is still nowhere near the all time great Roger Federer) winning over Michael Llodra after reaching deuce:

As an aside, I wonder the origin of the strange game scoring convention in tennis: 15, 30, 40 (was it 45 at some point and shortened to 40?). Many people colloquially shorten 14 to 5. I found some references to the history behind this strange scoring at http://www.quora.com/Tennis/…

Let us get back to our analysis and pretend we start off from deuce and then analyze the odds of my winning the game. So, I can win the game in two points after deuce (with probability 1/3 *1/3 which is equal to 1/9) or I can win the game in four points after a deuce by going back to deuce in between and then winning two points in a row. The probability that we will get to a deuce starting from a deuce in two points is:

4/9 = 1/3X2/3 (i win the first point and opponent wins the next point) + 2/3X1/3 (opponent wins the first point and I win the next point).

In theory there can be one deuce, two deuces, three deuces, or as many deuces as players can play and the probability of each successive deuce is 4/9 times the probability of the previous deuce. And as we calculated before, the probability of a win for me, starting from a deuce in just two points is 1/9.Therefore, the probability of an eventual win for me starting from a deuce is the probability I win in two points, or in four points (after one more deuce), or in six points, (after two more deuces) and so on.

This is nothing but the infinite geometric series 1/9(1 + 4/9 + (4/9)^2 + (4/9)^3 + …) which is 1/9(1/ (1-4/9)). You may recollect that the sum of an infinite geometric series 1+r + r^2+.. with a constant ratio r (r < 1) is 1/1-r. Simplifying the above sum we get 1/5 as the probability of me winning a game once we have reached deuce. It is clear that 1/5 is an upper bound on the chances that I can win a real tennis game (given that I am the inferior player) which is quite different from the 1/3 we started off for a single point. In any case, it is apparent that small differences get amplified over the long term and the gap widens.

I will not do the math, here in this post, for the real game according to normal rules, but it turns out that the probability of me winning the game turns out to be closer to 1/7 or so.

But hidden in my tennis analysis is a great life lesson. There is great value in making small incremental changes, and improving things little by little, and repeating this over a long time. The gap between people/companies who do this consistently over a long time widens. So, the key to building a great company or living a great life is to figure out a way for surviving the next day and improving on the things that matter (such as, one tennis point, one pixel on the user interface, one customer interaction or one sale, at a time)!!