Category Archives: Asked & Answered

Asked & Answered 7.0

You are the first-year coach of the Texas Lady Longhaulers of the WNBA.  This morning finds you dispirited after an embarrassing 88-60 loss to the Nashville Wynettes last night.  You need a more productive starting lineup for tomorrow’s game.  Where to begin?

Point guard is your most important position and your roster choices are Maya Thomas and Tamika DeShields.  They are good but very different players.  Maya is a steady shooter who has a 50% chance of sinking each shot.  Tamika, on the other hand, is streaky: if she sinks a shot, she makes her next one 80% of the time; but if she misses, she tends to miss again, 80% of the time.

So which point guard would you start, Steady Maya or Streaky Tamika?

I thought I would make this edition of Asked and Answered more interactive than usual. Readers, before I begin the discussion, you are invited to weigh in with your own answer. On the form below, check the box for either Steady, Streaky, no difference, or it depends.  There are no penalties for incorrect answers so just go ahead and check the box that your first-year-coaching gut tells you is the best one.  Then click the Vote button to record your answer and view the totals so far.

Coming Soon
The Better Player: Steady or Streaky?
The Better Player: Steady or Streaky?
The Better Player: Steady or Streaky?

First a little table-setting.  Maya and Tamika are fictional stand-ins.  Top players in the WNBA make 15-20 field-goal (two-point) attempts per game and sink 45-50% of those.  Maya’s performance is realistic, but whether Tamika’s streakiness is seen in actual players is a question for sports statisticians to answer.

Maya’s expected performance is easy to calculate.  Assuming she makes 20 attempts and sinks 50% of them, you may expect 10 field goals from her in a typical game, give or take.  The probability P(n) of Maya scoring exactly k field goals in n shots is given by

P(n) = \frac{n!}{k! (n-k)!} \;p^{k} q^{n-k}

where p is the probability of making any given shot and q is the probability of missing it.  [This is the well-known formula for the binomial probability distribution.  Exclamation points denote the factorial operation — they do not express my surprise.]  So the chance that Maya will score 10 +/- 1 field goals is about 50 percent:

Chance of 9 scores 0.160
Chance of 10 scores 0.176
Chance of 11 scores 0.160
Total 0.497

Predicting what to expect from Tamika is more complicated.  As I learned while studying this problem, her performance is an example of a Markov chain.  This is best explained by the diagram below.  Tamika starts the game in the Initial box.  After taking her first shot, she moves either to the Scored box (blue) or to the Missed box (violet).  The number next to each path shows the chance that she will follow that path when she shoots.

Every time Tamika shoots, she moves along a path.  Some paths return to the same box.  For instance, when Tamika is in the Scored box (i.e., she sank her last shot), she has an 80% chance (0.8) of circling back to the Scored box with her next shot.  Otherwise she moves over to the Missed box.  And so on.

No paths lead to the Initial box because, when taking a shot, Tamika only scores or misses. In my intro, I failed to specify how Tamika typically performs on her first shot.  For now, assume that Tamika starts the game cold as if she has just missed.  This implies that her chance of moving from Initial to Missed is 80% and from Initial to Scored is 20%.

The number of times Tamika visits the Scored box gives us the number of field goals you may expect her to score during the game.  But how do you calculate that?  Luckily, thanks to folks like David L. Deever, professor emeritus of Otterbein University in Ohio, there are such things as Markov chain calculators.  You enter the path information into a table, and the calculator returns the probability that a given box is occupied on the nth step.

Here are the results Dr. Deever’s calculator produced for Tamika.  The Scores column shows the expected number of times Tamika has scored after taking n shots:

The table shows that, after 20 shots, Tamika’s expected number of scores is 9.25, which is 0.75 less than the 10 scores you can expect from Maya.  Tamika never fully recovers from her cold start, and she will trail Maya by 0.75 scores (on average) forever.

So, if you had decided to start Steady Maya, your answer would be correct.

But perhaps you assumed Tamika has a 50/50 chance of scoring/missing on her first shot, after which the 80/20 rule would apply.  If that were the case, then Tamika would not fall behind Maya at all.  Each player would score 10 times in 20 shots, on average.  So if you guessed there would be no difference, your answer would also be correct.

• • • 

Tomorrow has turned into today, and your team is in the final minutes of its next game.  Since Maya had a slight edge over Tamika, you decided to start her today.  Unfortunately, your Lady Longhaulers have allowed too many turnovers, and they have fallen behind by 10 points.  You figure you will get five more possessions before the final buzzer.  To have any chance to win, your team will need to score on every one of those possessions.

You call a time-out.  Do you stay with Steady Maya or do you send in Streaky Tamika?

This gets interesting.  You need a player to sink five shots in a row.  The chance that Maya can do this is 0.5 (the probability of her sinking any one shot) to the fifth power, or 3.1%. The chance that Tamika can do it, coming in cold, is 0.2 (her first-shot success rate) times 0.8 (her repeat success rate) to the fourth power, or 8.2%.  Tamika is more than twice as likely to tie the score than Maya, though her chances are still slim.

So if your answer was to send in Streaky Tamika, you would be correct.  And this means that if your original answer was “it depends” then you would also be correct.  Now, all Tamika has to do is sink her first shot.  And the next.  And the next…

With that, your time-out is over.  May you enjoy the final minutes of your coaching career.

• • • 

David L. Deever, the author of journal articles as well as the Markov calculator that I used,  taught his last mathematics class at Otterbein University in 2003, ending a 37-year career.  His Facebook page (which has not been updated since 2013) reveals Dr. Deever to be a kind person, concerned citizen and a liberal in good standing.  I thank Dr. Deever for his contributions and I wish him good health.

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Asked & Answered 6.0

Blame it on a simple twist of Hickenlooper fate.

I confess: it is only because former Colorado Gov. John Hickenlooper decided to run for president that I decided to write this post.  His candidacy intrigues me — if Hickenlooper were to win, he would would join Dwight Eisenhower as our only four-syllable presidents and he would be the first president with a 12-letter surname.

Hickenlooper himself talks about his “funny name” but how unusual is it, relative to other U.S. Presidents?  How does it affect his chances?  What is a presidential name, anyway? Since no one is addressing these important questions, I am happy to do so — not only for Hickenlooper’s sake but for all of the 2020 presidential candidates.

Here is how we will proceed.  First, we will assign weights to the names of our presidents based on their positive or negative influence.  We will then use those weights to tabulate the desirability of three properties of presidential names — length, consonant-vowel ratio and last letter.  Finally, we will calculate a Sounds Like A President (SLAP) score for each candidate’s name based on its properties and their historical desirability.

• • • 

Let’s preface the discussion by comparing the length of the surnames of U.S. presidents to those of the population at large.  The chart below shows the prevalence of last names of various lengths for both groups.[1]  Note how the presidents with 8 to 10 letters in their last names are over-represented with respect to the population:

We should not conclude from this chart that long names confer an electoral advantage.  Perhaps the performance of those presidents was so poor that a longer name now carries a negative connotation.  Or it could be that no one even remembers certain presidents and therefore the characteristics of their names are irrelevant.  This is why we must begin the analysis by calculating Presidential Name Weights (PNW):

PNW (for President x) = Reputation Score  x  Memorability Factor

Using Presidential Name Weights (ranging from -1.0 to 1.0) is more realistic than giving equal weight to each name without regard to how or whether a president is remembered.

The Reputation Score of each president is derived from the 2019 Survey of U.S. Presidents conducted by Siena College Research Institute.[2]  In this survey, 157 presidential scholars and historians rated the presidents on their abilities and accomplishments, and the results were ranked from 1 (best) to 44 (worst).  I re-scaled the rankings so that Reputation Score ranges from 1.0 (best) to -1.0 (worst) and 0.0 represents an average (mediocre) president.

The Memorability Factor for each president comes from a study by Roediger and Desoto, in which participants were asked to name as many presidents as they could remember.  This factor ranges from 0.0 to 1.0, reflecting the fraction of participants who were able to name a given president.[3]

We can now calculate the Presidential Name Weight for each president as the product of his reputation and his memorability.  The results (below) are sorted from most positive to most negative:

This chart tells us that the names Washington, Lincoln, Jefferson, Roosevelt and Kennedy carry the most positive weights, whereas Nixon, Hoover, Johnson, Bush and Trump suffer the most negative weights.  Candidates whose names remind us of Washington or Lincoln would score highly relative to candidates with names like Bush and Trump.

• • • 

Our next step is to construct a Desirability Table for each property of a presidential name.  For instance, how desirable is having 9 letters in one’s last name?  How desirable is having a name that ends in n?  And so on.

This example shows how a Desirability Table is built.  Say we want to know the desirability of having a surname with x letters.  1) We list all the presidents with x-letter surnames and add up their Presidential Name Weights.  2) We divide that sum by the total PNW for all presidents to yield the desirability factor D for an x-letter surname.  Repeat these steps for every possible value of x to complete the Letter Count Desirability Table.  And so on.

Here then are the Desirability Tables for the presidential name properties in our analysis.  Each D-factor is a value between -1.0 and 1.0, and the highest D-factor for each property is shown in green.

The tables reveal that the highest-scoring presidential name would be 9 letters long, have a consonant-to-vowel ratio between 2.1 and 2.6, and end in the letter n.  Interestingly, none of our presidents’ names have all three of these features.[4]

The last item we need to address before calculating Sounds Like A President (SLAP) scores for the candidates is the weight we should assign to each property.  Here, we will elect to base a property’s weight on the amount of variation in its D-factors.[5]  Sparing the reader my lengthy justification, the weights we will assign to letter count, consonant-vowel ratio and last letter are w(LC) = 0.265, w(CV) = 0.275 and w(LL) = 0.460 respectively. 

• • • 

We are finally ready to calculate the SLAP score for each candidate, using this formula:

SLAP (for candidate x) = w(LC) x D(LC)  +  w(CV) x D(CV)  +  w(LL) x D(LL)

For illustration, let’s consider John Hickenlooper.  Hickenlooper’s last name has 12 letters, a consonant-vowel ratio of 1.4 and ends in r, so his SLAP score is

SLAP (Hickenlooper) = (0.265 x 0) + (0.275 x 0.463) + (0.460 x -0.038) = 0.110

which means he does not have a very presidential name, indeed.

The highest possible SLAP score is 0.657, corresponding to a name which has 9 letters and 5 consonants and ends in n.  Sen. Dianne Feinstein, destiny awaits you.

• • •

The table below lists the SLAP scores for the 20 mainstream presidential candidates (plus Donald Trump).  And the name of our next president is…

Elizabeth Warren!  Warren vaults to the top of the list on the strength of her last letter n and consonant-vowel ratio of 2.0.  Warren edges out Joe Biden, whose five-letter name weighs him down, thanks to the likes of Trump, Nixon, Tyler, Hayes and Grant.

Tim Ryan of Ohio has a decent SLAP score and may be a good running mate for Warren. Bernie Sanders has another respectable showing but comes up short in the last letter race. Peter Paul Montgomery Buttigieg (his full name) is just happy his last name is not Burns.

If only Hillary Clinton were running… her Sounds Like A President score would be 0.637.

Last and least is where we find Donald Trump.  Memorably poor performance.  Negative scores in all areas.  Still doesn’t sound like a president.


[1] Data for the U.S. population is from the 2010 U.S. Census, comprising the most-common 150,000 names representing 90% of the population.

[2] As one might expect, George Washington topped the rankings and Andrew Johnson came in dead last.  Donald Trump was merely the third-worst.

[3] The Roediger and Desoto study was published in 2014, prior to the 2016 election.  As such, the president named most often was Barack Obama (100%).  I assigned Donald Trump a 100% memorability factor as well but did not adjust the figures for the other presidents.

[4] Readers may ask, why these properties and not others, such as the number of syllables?  Syllable count would be fairly redundant, as it is correlated with letter count and consonant-to-vowel ratio.  I tried to select properties that appeared to be more-or-less independent but I did not actually test their cross-correlation.

[5] Specifically, the property weights w are based on the standard deviation in D(property).  My rationale is that a factor with little variation does not differentiate the candidates as much as one with large variation.  The statisticians among my readers are sure to howl, which is OK, because I don’t know any statisticians.

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Asked and Answered 5.0

I recently became a bitters-and-club-soda drinker.  As I was to discover, club soda has the annoying habit of going flat before one finishes the bottle.  To reduce waste, one can buy small bottles of national brands but they are much more expensive than the one-liter or two-liter bottles of store brands.  What’s a club soda drinker to do?  Glad you asked, and here’s your answer.

If club soda did not go flat, the choice would be clear.  A six-pack of 10-ounce glass bottles of White Rock Club Soda costs $3.99 here — 33.25 cents per 5-ounce serving — while four one-liter bottles (135.3 ounces) of our store-brand club soda cost only $3.00 — 11.09 cents per serving.  This means club soda in the 10-ounce bottles is three times more expensive… at least before any drinks are poured.

I tried a few one-liter bottles and was surprised to see how much club soda went to waste.  (Soda without sparkle belongs in the drain, not the drink.)  So I sat down to figure out whether the one-liter variety was a bargain or a boondoggle.  The answer would involve Henry’s Law, the Ideal Gas Law, bottling industry data and a few simplifying assumptions. (Full details are provided in the Appendix.)

But let’s start with how club soda goes flat (Figure 1).  The icon at far left represents the bottle as you buy it — filled to the neck with liquid and then topped off with high-pressure carbon dioxide (CO2).  Then, you open the bottle, pour a serving, and re-close the bottle.  The headspace is now filled with air from your room.  Over time, as the bottle rests in your refrigerator, CO2 escapes from the soda and collects in the headspace, until the pressure reaches an equilibrium (as determined by Henry’s solubility constant for carbon dioxide).  This equilibrium CO2 pressure will be lower than the original pressure — as suggested by the lighter shades of blue in the diagram — due to the CO2 that escaped when you opened the bottle along with the CO2 that you poured into your drink.

FIGURE 1: How Club Soda Goes Flat

The club soda gets flatter each time you open the bottle and pour a serving, until hardly any gas is left.  At some point, you may as well water your plants with what remains.

The question is, for a given serving size and bottle size, how many servings can one pour before the contents become undrinkable?  In my case, a serving is 5 oz (148 ml), I drink one serving a day, and I consider club soda too flat to drink if it has less than half of the original fizz (your mileage may vary).  This is the case I will address first, comparing the 10-ounce bottles to the one-liter bottles.

Let’s assume we refrigerate the bottle at 4C (39F).  Once a day, we take it out of the fridge, open it, pour the 5-oz serving, then close the bottle and return it to the fridge, all done as quickly as possible so that the contents do not get a chance to warm up.  We repeat these steps every day until the bottle is empty or the soda is flat.  The following chart (Figure 2) shows the resulting carbonation level for each serving, for both sizes of bottles:

FIGURE 2: Carbonation Level vs Serving Number (One 5-Oz Serving per Day)

We see that the 10-ounce bottle delivers two acceptable servings — the carbonation level of the second serving (57%) satisfies my 50% specification.  But the real surprise is that the one-liter bottle yields just one more drinkable (61%) serving — the fourth serving has only 36% of the original fizz and the fifth 17%.

So, by my criterion, almost 56% of the one-liter bottle is undrinkable.  This increases the effective cost-per-serving of a one-liter bottle from 11 cents to 25 cents.  Even so, it is still less expensive than the 33 cents-per-serving cost of a 10-ounce bottle.

Pouring two servings at a time (1o ounces) makes the one-liter bottle an even better value, but not by much.  The first two servings would be full strength of course, and the next two would have 72% of the original fizz.  The final two servings, however, would drop to 34%. The effective cost-per-serving in this scenario would be 19 cents, and 41% of the contents would still be undrinkable.

To completely eliminate waste from a one-liter bottle, you would have to either (a) lower your fizz-level standards, or (b) belt down half the bottle (3-plus servings) each time that you open it.  Only then will it cost you 11 cents per serving.

The bottom line is, a one-liter bottle of club soda at $0.75 is a better buy than a six-pack of glass bottles at $3.99, even if you have to share half of it with your houseplants.  But if the one-liter bottle costs $1.00, it is a break-even proposition.

Keep the Fizz Alive

Want to get the most fizz for your club-soda buck?  Research by Nestlé [1] suggests that soda in one-liter plastic (PET) bottles loses about 10% of its carbonation every 60 days when stored at room temperature.  This is because CO2 gas diffuses through the walls of the bottle.  (Bottlers compensate for the loss in shelf-life by adding extra carbonation to PET bottles.)  There are three takeaways from this:

(1) Buy your club soda from a busy supermarket.  This increases your chance of drinking  recently-bottled higher-carbonation product.

(2) Only buy as many bottles as you will use this week.  For the same reason.

(3) Refrigerate your club soda.  This reduces the internal pressure — and thus the rate of CO2 loss due to diffusion — by about one-third compared to room-temperature storage.  Not only that, cold liquid dissolves more CO2 so less gas is lost when you open the bottle.

Follow this advice and you will earn an A for effervescence.  And don’t forget to recycle.


To estimate the carbonation in a bottle of club soda at each step of its product life, I made a number of reasonable simplifying assumptions:

  • The bottle is completely rigid.
  • The gases and liquids follow Henry’s Law and the Ideal Gas Law.
  • When the bottle is filled, the headspace gas is assumed to be pure CO2 .
  • The carbonic acid formed by the reaction of CO2 and water is insignificant.
  • The diffusion of CO2 through the walls of the PET bottle is ignored.
  • Evaporation of the soda is insignificant.
  • The liquid and gas are at equilibrium before the bottle is opened.
  • Bubbles that escape from the soda just before it is poured are ignored.
  • The contents of the bottle remain at 4C (39F) while the soda is poured.
  • When the soda is poured, the gas in the headspace is replaced entirely by air.
  • The amount of CO2 and water vapor in the air can be ignored.

We begin by writing down the initial conditions in each bottle (Figure 3).  I will use liters as the unit of volume, atmospheres as the unit of pressure, and moles for the mass of CO2. (One atmosphere is essentially sea-level pressure, and one mole of CO2 is about 44 grams.)

FIGURE 3: Initial Conditions

A few notes on these facts and figures.  I measured the total volume of each bottle in my kitchen by filling them to the brim with water.  I assumed that the actual amount of soda in each bottle was exactly what the label claimed (bottling equipment is very accurate).  The initial headspace is the difference between those numbers.

According to Steen and Ashurst [2], bottle-filling is usually performed at 14C (about 57F) and the carbonation pressure for club soda is about 60 psi, or just over 4 atmospheres [3].  Assuming that all the gas in the headspace is carbon dioxide, we can use the Ideal Gas Law and Henry’s constant to calculate the amount of CO2 in the gas and liquid as bottled.

Initial moles CO2 in headspace (n0) from the Ideal Gas Law [4]:

\text{(1)}\hspace{5em}n_0 = p_0V_0 / RT_0

where p0 is the CO2 pressure (4.083 atm), V0 is the initial headspace volume (0.010 liters or 0.85 liters depending on bottle size), R is the gas constant (0.08206 liter-atm/mol-deg) and T0 is the filling temperature in degrees Kelvin (14C + 273.15 ≈ 287K).

Initial concentration of CO2 in liquid (c0) from Henry’s Law [5]:

\text{(2)}\hspace{5em}c_0 = p_0H_{14\text{C}}

where H14c is Henry’s constant at 14C (refer to above table). 

We now know the total moles (m0) of CO2 in the bottle:

\text{(3)}\hspace{5em} m_0 = n_0 + c_0L_0

where L0 is the initial liquid volume (0.296 or 1.000 liters depending on bottle size).

Chilling the bottle to 4C does not change the moles of CO2 in the bottle but it does affect the CO2 liquid-to-gas ratio.  We find the new liquid concentration (c1) at 4C by combining the Ideal Gas Law and Henry’s Law, then substituting for pressure p1 and rearranging:

\text{(4)}\hspace{5em} n_1 = p_1V_1/RT_1 = m_1 - c_1L_1\;,

\text{(5)}\hspace{5em} p_1 = c_1/H_{4\text{C}}\;,\hspace{.75em}\text{and so...}

\text{(6)}\hspace{5em} c_1 = m_1H_{4\text{C}}/(L_1H_{4\text{C}}+V_1/RT_1)

where m1m0 L1 = L0 and V1 = V0 (since nothing has been removed from the bottle).  Knowing ci lets us back-calculate n1 and p1 from Equations (4) and (5).

We now pour the first serving of soda, whose volume is LS .  The pour reduces the liquid in the bottle to L2 = L1Land the total moles of CO2 to m2 = m1n1c1 LS , as we assume all CO2 in the headspace is lost.  The headspace volume is now V2 = V1 + LS .  We quickly close the bottle and return it to the refrigerator, so that T2T1 .  While the bottle rests, the CO2 concentration of the liquid gradually decreases from c1 to cand the CO2 gas pressure in the headspace rises to p2 .  The values of c2 and p2 are unknown, but we can solve for them using Equations (5) and (6) with new subscripts:

\text{(5')}\hspace{5em} p_2 = c_2/H_{4\text{C}}\;,\hspace{.75em}\text{and...}

\text{(6')}\hspace{5em} c_2 = m_2H_{4\text{C}}/(L_2H_{4\text{C}}+V_2/RT_2)

This is the procedure I used to generate the CO2 concentration results in this article.


[1] Profaizer, Mauro. “Shelf life of PET bottles estimated via a finite elements method simulation of carbon dioxide and oxygen permeability.” Italian Food and Beverage Technology, vol. 48, 2007.

[2] Steen, David P., and Ashurst, Philip R. (editors).  Carbonated soft drinks: formulation and manufacture.  Oxford Ames, Iowa: Blackwell Publishers, 2006.

[3] Spangenberg, Craig. “Exploding Bottles.”  Ohio State Law Journal, vol. 24, 1963, p. 513.

[4] Khan Academy ( is one of thousands of sites with info on the Ideal Gas Law.  The Ideal Gas Law (pV=nRT) is to chemistry what Newton’s Second Law (F=ma) is to physics.

[5] Choose your own reference: Henry’s Law on Wikipedia or Henry’s Sparkling Water on ice.

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