**Asked & Answered 7.0**

**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.

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 *n*th 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.