Tennis and best 2-out-of-3 vs. best 3-out-of-5

I recently read a very interesting article on regarding men’s and women’s tennis that reminded me of the following standard problem in probability.

Player X and Player Y play a series of at most n games, and a winner is declared when either Player X or Player Y wins at least n/2 games. Suppose that the chance that Player X wins is p, and suppose that the outcomes of the games are independent. Find the probability that Player Y wins if (a) n = 3, (b) n = 5.

The easiest way to solve this is to assume that all n games are played, even if that doesn’t actually happen in real life. Then, for part (a), we can use the binomial distribution to find

  • P(X = 0) = P(Y = 3) = (1-p)^3
  • P(X = 1) = P(Y = 2) = 3p(1-p)^2
  • P(X = 2) = P(Y = 1) = 3p^2(1-p)
  • P(X = 3) = P(Y = 0) = p^3

Adding the first two probabilities, the chance that Player Y wins is (1-p)^3 + 3p(1-p)^2 = (1-p)^2 (1+2p).

Similarly, for part (b),

  • P(X = 0) = P(Y = 5) = (1-p)^5
  • P(X = 1) = P(Y = 4) = 5 p (1-p)^4
  • P(X = 2) = P(Y = 3) = 10p^2 (1-p)^3
  • P(X = 3) = P(Y = 2) = 10 p^3 (1-p)^2
  • P(X = 4) = P(Y = 1) = 5 p^4 (1-p)
  • P(X = 5) = P(Y = 0) = p^5

Adding the first three probabilities, the chance that Player Y wins is (1-p)^5 + 5p(1-p)^4 + 10p^2(1-p)^3 = (1-p)^3 (1+3p+6p^2).

The graphs of (1-p)^2 (1+2p) and (1-p)^3 (1+3p+6p^2) on the interval 0.7 \le p \le 0.9 are shown below in blue and orange, respectively. The lesson is clear: if p > 0.5, then clearly the chance that Player Y wins is less than 50%. However, Player Y’s chances of upsetting Player X are greater if they play a best 2-out-of-3 series instead of a best 3-out-of-5 series.

best2outof3Remarkably, this above curve has been observed in real-life sports: namely, women’s tennis (which plays best 2 sets out of 3 — marked WTA below) and men’s tennis (which plays best 3 sets out of 5 in Grand Slams — marked ATP below). The chart indicates that when two men’s players ranked 20 places apart play each other in Grand Slams, an upset occurs about 13% of the time. However, the upset percentage is only 5% in women’s tennis. (This approximately matches the above curve near p = 0.8.)

However, in tennis tournaments that are not Grand Slams, men’s tennis players also play a matches with a maximum of 3 sets. In those tournaments, the chances of an upset are approximately equal in both men’s tennis and women’s tennis.

However, since the casual tennis fan (like me) only tunes into the Grand Slams but not other tennis matches, this fact is not widely known — which gives the misleading impression that top women’s tennis players are not as tough, skilled, etc. as men’s tennis players.

The FiveThirtyEight article argues that top women’s tennis players don’t make it to the latter stages of Grand Slam tournaments than top men’s players because of the two tournaments are held under these different rules, and that women’s tennis would be better served if their matches were also played in a best-3-out-of-5 format.



On Cheating

Here’s an article on cheating that I read recently:

A couple of quotes:

It was once a common schoolyard taunt, delivered with a snarling, singsong cadence: “Cheaters never PROSper! Cheaters never PROSper!” The perp, having been caught in the act of copying from a neighbor’s paper, would reflexively hiss: “I am not a cheater!”

A rash of cheating scandals at leading universities has again aimed a spotlight on the issue and renewed interest in understanding what motivates people to flout the rules. Researchers have found that even excellent students can be tempted to cheat when certain conditions line up. When we’re feeling tired or overwhelmed; when the likelihood of getting away with it is high; and when there is a perception that others are cheating, too, it can be easy to rationalize taking the low road.

Monin says two conditions clearly can motivate cheating among high-achieving students: first, the perception that “everybody else is cheating, so it can’t be that bad;” second, the worry that “I’ll fall behind unless I cheat.” (The same reasoning was used to explain steroid use by baseball players and blood doping by professional cyclists.) The psychological underpinnings of cheating are important, but administrators, researchers and ethicists are trying to understand whether other factors—technology, parenting styles and pressure-cooker environments on campuses—are driving more cheating today.


Duke University researcher Dan Ariely is a professor of behavioral economics who wrote the 2012 book The (Honest) Truth About Dishonesty. Ariely notes that for many years the dominant school of thought about cheating followed the theories of Nobel Prize-winning economist Gary Becker. Becker argued that people rationally analyze their decisions, weigh the pros and cons, assess the risks and penalties, and land on a choice that produces the best outcome for the lowest cost.

Ariely decided to test that. He created experiments involving math problems in which subjects were tempted by greater or lesser rewards, and higher and lower chances of getting caught, and in which the behavior of others around them informed their mindset. Based on his results in one experiment, 70 percent were tempted to cheat at least a little. But it turns out the value of the reward plays a lesser role than Becker would have predicted. Instead, people seemed to calculate the largest benefit they could get for a transgression that would still allow them to tell themselves they weren’t a cheater. “Essentially, we cheat up to the level that allows us to retain our self-image as reasonably honest individuals,” Ariely says.

I recommend reading the whole article.


Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}},

\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}},

\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}.

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

Error involving countable numbers in Glencoe Algebra 2 (2014)

Errors in textbooks happened when Pebbles Flintstone and Bamm-Bamm Rubble attended Flintstone Elementary, and they still happen on occasion today. But even with that historical perspective, this howler is a doozy.

This was sent to me by a former student of mine. It appears in the chapter study guide for Section 2.1 of Glencoe’s Algebra 2 textbook (published in 2014), presumably as an enrichment activity for students learning about the definitions of “one to one functions” and “onto functions.”

countable infinityJust how bad is this mistake?

  • The above “proof” is only a blatant assertion, without any justification, either formal or informal, for why the authors think that the statement is false.
  • The ordering of the rational numbers in the way listed above is actually reasonably close to the listing that actually does produce the one-to-one correspondence between \mathbb{Q} and \mathbb{Z}.
  • Just above Example 2 was Example 1, which gives the correct proof that there’s a one-to-one correspondence between \mathbb{Z} and \mathbb{N}. If the authors had double-checked this proof in any reputable book, they should have also been able to double-check that their Example 2 was completely false.

countable infinity 2

green line

The full chapter study guide can be found here (it’s on the last page):

Reactions can be found here:

Reference to this can be seen on page 10 of the teacher’s manual here:


The Shortest Known Paper Published in a Serious Math Journal


Euler’s conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years. Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences. Their article — which is now open access and can be downloaded here — appeared in the Bulletin of the American Mathematical Society.