Monthly Archives: November 2019

Last month, my spouse and I made a 13-day trek to several western states: Colorado, Utah, New Mexico, Arizona and California.  Our trip served multiple purposes — see old friends (some for the first time in decades), visit family, get a better sense of the West, and enjoy some good food and fine art.

While we spent a few days on our own in Santa Fe, our trip would not have been the same without (more than) a little help from our friends.  It was certainly great to see them and share experiences, but it was also a privilege to have such well-versed guides escorting us to the places we visited.  Relying on them allowed me to make less-detailed travel plans than I normally do, which heightened our sense of discovery.  I appreciate all their time and effort, as well as all the what-to-see-and-do suggestions that many others offered.

I took 1000-plus photos, some of which had to be good if only by accident.  I have posted the more artsy ones at my other site (ART@CHC) in two newly-minted galleries: Western Skies (for landscapes) and Western Civilization (anti landscapes?)  Interesting photos from this trip that didn’t fit either category may one day find their way into future and/or existing galleries. But I’ll share a few of them now.

For example, this bighorn sheep (click image to enlarge) strolled right past our car window as we were parked on a roadside in Colorado National Monument.  Such lucky encounters with natural beauty can turn anyone with a camera into Ansel Adams for a small moment.  But sometimes it’s better to forget the camera and remember to just be there.

After a few days exploring Grand Junction with our friends, we all drove the river road to Arches National Park near Moab, Utah.  The river road (Utah 128) is impressive in its own right, following the Colorado River as it flows past the gorges it created.  The most dramatic views were those of Fisher Towers. As my inner Keats puts it:  The morning light bath’d its cragged spires in mystery.

Many of the rockfaces along the gorge were spectacular as well, but they were no match for what we saw in Arches.  We only had a morning to spend in the park, so we mostly stayed on the loop road — but there was still plenty to take in.  My main takeaway was how varied the terrain was — not just the pillars of rock sculpted in a hundred different ways and all the boulders balanced on top of other boulders, but the ground itself: sandy in one place, gravelly in another, nearly lava-like elsewhere.  I could have spent hours walking along the Arches trails and shooting at ground level.

The internet has plenty of great photos of Arches by great photographers, so I’m not going to duplicate them here.  I did want to point out one rockface that I saw as we were driving the loop road — this rockface actually did have a face and, not only that, it was rocky:

Rockface Balboa (Arches National Park) by CHCollins

We drove from Utah to New Mexico and spent four days in the Santa Fe area.  I grew tired of turquoise pretty quickly but not so the art:  there were dozens of interesting galleries on Canyon Road, along with the Georgia O’Keeffe museum and New Mexico Museum of Art.  I was surprised and pleased that the O’Keeffe museum was not preoccupied with paintings of sexually-suggestive flower parts.  Most of her work was far less notorious.  (Incidentally, O’Keeffe denied those connotations, which I find hard to believe.)

If there was one O’Keeffe painting that I would have brought home, it would be this abstract waterfall (right).  That’s the kind of art I wish I could imagine and do.

I did bring home a souvenir of sorts from one Canyon Road gallery, namely a small rhodochrosite (“Rose of the Inca”) warbler perched on a veined rock of unknown type.  Even after the owner dropped the price by a third, I probably paid more for it than it was worth.  But it’s a nicer remembrance than a Georgia O’Keeffe coffee mug would have been, so no regrets here.  I am admiring it as I write.

Leaving Santa Fe, we took the Turquoise Trail (State Route 14) to Albuquerque, stopping at several points of interest along the way.  Our first stop was the San Marcos Cafe and Feed Store, where we had a great breakfast watching chickens and peacocks and listening to a well-traveled guitarist-singer.  It was a quirky spot, one of those hidden gems.  I just had to order the Eggs San Marcos, an egg burrito with chiles, cheese, guacamole and beans.  This was the only burrito (or variant thereof) I ate the entire trip and it was delicious.

Our second stop was the village of Madrid, which was frankly a disappointment.  It was essentially a half-mile stretch of chuchería (tchotchke) galleries, a chocolate shop with a dauntingly long line (sorry, not going to wait 15 minutes for chocolate) and a biker bar.  Almost impossible to find a space to park that Saturday afternoon.  Maybe I’m not being charitable toward the place, but I wasn’t into that vibe that day.

Our last stop on the Turquoise Trail was the roadside haunt of one Mr. Leroy Gonzales of Golden, New Mexico.  His abode has a pin on Google Maps titled “Bottle House” but it is much more than that.  We didn’t even know the house or Mr. Gonzales existed until my spouse spied the bottles in front of the house and made a U-turn so we could check it out.

My brief visit with Mayor Leroy (see my Western Civilization gallery) was the highlight of my trip down the Turquoise Trail.  Leroy and his domain have been well documented by many others, including biker/photographer Chuck Jines, so I will just provide a link and let Leroy do the talking.  Enjoy.

• • •

We visited family in San Diego for a few days, then headed eastward on Interstate 8, eventually driving through the strangest hills I’ve ever seen.  As I would later learn, these are the Jacumba Mountains — they look exactly like giant piles of rubble, as if they were just dumped there from some astronomical-scale construction project.

Other notable features of Interstate 8 included views of the Trump Wall rising and falling amidst the sand dunes, and interior border-control checkpoints.  I didn’t even know that the latter existed until this trip.  I am sure my spouse and I were being ethnically-profiled by those smiling agents, just as when we visited Canada back in the day.  Our freedom of movement at such checkpoints is merely the visible tip of our white-privilege iceberg.

The final leg of our trip included a four-legged breakfast ride at a resort ranch near Tuscon, Arizona.  I was assigned a draft horse whose chest was so big that the saddle wouldn’t stay on straight.  Halfway through the ride, my saddle starting slipping to the left and my only choice was to fall off as gracefully as I could without getting my foot caught in the stirrup.  This earned me a lecture from the ride-boss to sit straight and push on my stirrups if the saddle began to slide — which it did, again.

Meanwhile, my spouse did not anticipate that the trail ride would include rock-strewn 30-degree downslopes.  Neither did I, but I figured that my horse had learned the trail on previous rides, and I was pretty sure he didn’t want to fall either.  So I let him do his thing and we all lived to tell about it — which I am doing now.

One last thing: I usually don’t do this, but here is a link to a set of uncropped, unadjusted, uncurated and unexplained straight-from-the-camera shots from our Mountain Time trip.  (Don’t forget about the nicely-processed images in ART@CHC).  In any case, thanks for saddling up with me, and have a great Thanksgiving.

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Dyslexic Driving

Photos and digital graphics by  Original road sign symbols by Federal Highway Administration.
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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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