The Monty Hall Problem

 “It’s easier to fool people than convince them they’ve been fooled.” – tenuously attributed to Mark Twain

Anchor bias: the tendency in decision-making and related cognitive processes to unduly emphasize information obtained at a certain point in time over any newly-discovered information. This bias tends to take the form of sticking with an initial position, and is likely informed by avoiding the mental effort and presumed risk of changing one’s position from that which is currently held.

Or something like that.

Note, much of the background information for today’s column was derived from a Wikipedia article on the subject. You could read that, but this dispatch is way funner!

The Monty Hall Problem was first posed in American Statistician by one Steve Selvin in 1975, and later in a 1990 magazine column by Dr. Marilyn vos Savant, great-granddaughter of the famed idiot Cletus Savant. JK about that last part, but she is descended from noted 19^th and early 20^th century physicist Ernst Mach. Side note – you’ve really made it in science if your name is used in lowercase, i.e., a unit of measure is named after you. In this case, mach is the ratio of an object’s speed relative the speed of sound.

Marilyn vos Savant was, among other things, a rather brainy columnist in Parade Magazine.  Those of us with a degree of superannuation will remember the roto gravure magazines and ad flyers that accompanied the colored Sunday comics in the rainforest-killing newspapers of yesteryear. Parade Magazine was one such, and in it, vos Savant wrote “Ask Marilyn,” an intriguing Q&A column on brainy topics.

On September 9, 1990, vos Savant posed her most controversial column, on the Monty Hall Problem. It’s not the problem itself that’s controversial; it’s based on the (at the time) popular game show Let’s Make a Deal, a show with which most of her readers would be familiar, with only 3 or 4 VHF TV channels to choose from, and even worse pickings on UHF. The controversial part was the solution, and the rigorous and often contentious debate that followed.

First, let’s lay out the problem. The game show contestant is presented with three closed doors: A, B, and C. The contestant is informed that behind one door is A NEW CAR! Behind the others are goats.

A few assumptions are in order at this point:

1) The doors all look alike, are the same size, etc., i.e., not a garage door and two goat pen gates.

2) The car is not a Rivian and is therefore preferable to a goat.

3) The contestant is sufficiently urbane as to prefer the car over a goat.

4) The goats remain quiet and do not emit any telling odors.

And some game rules to remove any bias or other chicanery:

1) The a priori selection of which door conceals the car is assigned at random with a uniform probability distribution, i.e., it’s completely unbiased.

2) The assignment of door to car or goat does not change throughout the game, although the goats may discreetly trade places with each other.

3) Which door conceals which prize I s known to the host.

4) …and is not known to the contestant.

So much for the setup, now the game:

The contestant chooses a door, hoping it conceals a car. Monty Hall does not immediately open the door. Instead, he opens another door, revealing a goat. Remember, Monty knows where the goats and the car are. At this point, Monty offers the contestant the opportunity to change his or her initial choice to the other closed door.

The contestant then opts to stick with the original door choice or switch. Monty then dramatically opens the other doors to reveal the contestant's fate, car or goat, euphoria or despair, that's show biz! And speaking of show biz, I have no idea whether or not any contestant actually kept the goat. I know I would, and my next stop would be a halal butcher, but that’s beside the point.

So... what’s the problem?

The problem is, should the contestant stick with the initial choice, or, when given the opportunity, choose the other unopened door? Answer: switch!

vos Savant explained why in her famous column, and the controversy was that most of her readers, including some real brainiacs, didn’t believe it. That’s because, although we encounter hands-on applications of it on a daily, if not hourly basis, most people are not comfortable thinking consciously in probability space. But if today’s discussion goes well, you will be, at least more so than you are now.

I’m sure you can appreciate that the initial choice has a 1/3 probability of being correct. Therefore, the car has a 2/3 probability of being behind one of the unchosen doors. Then Monty supplies more information by revealing which of the unchosen doors conceals a goat. Maybe both unchosen doors conceal goats, or maybe just one did; in any case the contestant now sees one of the “wrong” choices.

So, of what value is this information? Note that information was added about the non-chosen doors, not the chosen one. The chosen door still has a 1/3 probability of being correct, but the new information assigns the 2/3 probability all to the unchosen but still closed door. 1/3 odds of winning by staying with the initial choice, 2/3 by switching. Therefore, switch!

Get it?

No? Consider this: instead of three doors, the contestant has 1000 from which to choose. Let’s say for simplicity that he or she picks door #1, and we can all agree that the odds of winning the car are 1/1000. The odds of the car being behind one of the unchosen doors is 999/1000, i.e., a near-certainty. Next, Monty opens up 998 doors revealing goats (the studio must be getting rather redolent by this time, and probably a bit noisy as well) – remaining closed are door #1 and door #637. Now, you and the contestant can see, of all the bad choices available earlier, only one (of the two closed doors) remains, and he or she would be exceedingly lucky not to have made one initially.

vos Savant’s column generated over 10,000 letters (that’s how people flamed before the internet), with only 8% of the respondents agreeing with her. Note, this does not mean that only 8% of readers agreed; probably more did, but would be less inspired to lick a stamp to make their case. The debate raged on for many months, in and out of the “Ask Marilyn” column, among academics and laypeople alike, accompanied with much acrimony and a side order of sexism. Numerous proofs and explanations failed to move the needle, and the controversy was ultimately resolved by schoolteachers across the US volunteering their class time to conduct simulations with their students (crowdsourcing before crowdsourcing was cool), empirically proving the point. In the end, only 70% (up from the initial 35%) of academics agreed with her. Not stated was what kind of academics – there may have been some Harvard faculty in the pool.

For today's little statistics and probability lesson, we'll let Clint Eastwood have the final word.

Upstairs Virginia

Hello and Happy New Year!

For an intellectually broadening experience, I’d like to share a private dispatch, with identifying info and specific location names obfuscated, otherwise as-is. What follows is an after-action report (AAR) for a Summits on the Air (SOTA) activation, which doubled as a Parks on the Air (POTA) activation, as requested by a good friend, to whom I will henceforth refer as BAC*, and with whom I normally do SOTA and POTA activations. BAC was unable to join me for these activations due to family obligations and he requested an AAR, as he was planning to activate the same summits/park the following weekend.

*…which stands for bitch-a$$ cracker, long story J

OK Bruck, what the Sam Hill are you talking about and why?

You’ve probably correctly deduced that “on the air” refers to radio communications, and in this case, ham radio. “Summits” are mountain peaks; SOTA is a program for tracking and logging (and collegially competing with) ham radio contacts from mountaintops. The program (or should I say programme) started out in England, hence the usage of “summits” for mountains. “Activation” means going up a designated mountain (there are scores of them within a day trip of Bruck’s and BAC’s home stations) and attempting to make radio contacts; a successful activation requires contacts with four or more unique stations.

POTA is essentially the same thing, but with state and national parks rather than mountain peaks, and one needs ten contacts for a successful activation. SOTA summits often reside in said parks, in which case SOTA activations can also be logged for POTA, providing that at least 10 contacts are made.

On 9 December, 2023, I activated Whistler and Smoke Bluff, in central Virginia for SOTA. These peaks are both within the Zinc Hill Wildlife Management Area (WMA), which is a designated POTA park, therefore I was able to combine the SOTA contacts for a POTA activation.

And just to forestall any misguided assumptions, Lucille is the Black F150 of Bruck (BFOB). Another illuminating note which will come in handy at the end – my photographer Ansel (not his real name) occasionally accompanies BAC and me on activation trips. On one such occasion, while BAC and I were activating a peak on a popular hiking trail, Ansel was chatting with a couple of young high school or college-age women. Henceforth we bestowed upon him the honorary callsign K4DOM, the suffix of which abbreviates dirty old man.

A couple of nomenclature notes for you: QSO means radio conversation, and S-S means summit-to-summit contact, i.e., contact with an operator doing SOTA on a different peak.

On to the next question: why?

Just thought I’d share some cultural insights into corners of the world that few people are aware of but exist all around us. And now, without further ado:

AAR for today's activations:

BLUF

Successful activations on Whistler and Smoke Bluff + made sufficient contacts for Zinc Hill WMA POTA activation. Did all SOTA activations on 2m FM and two POTA contacts on 10m SSB, one in Venezuela. Several S-S QSOs as there were a few other groups on the hills, mostly people we know.

DISCUSSION

The drive down and back were pretty long. I recommend that if/when we do it next year, we also tie in one or two other local summits to make it worth it. Or perhaps make it part of a camping trip. I worked the 10m contest a bit on the way down – made new friends in England, France, and Spain.

Riddell Rd. was NP for Lucille and would be manageable for any respectable 4wd vehicle, all the way to the top of Smoke Bluff. Most of it could be done in a regular SUV or perhaps a Subaru Forester, but I wouldn't take a normal car on it. Wouldn't be a problem for a mountain bike, aside from the effort. The road gets dicier as you approach Smoke Bluff, but still nothing like the Tuscarora Trail.

There's a yellow gate at the entrance to the WMA, prior to which either side of the road is private property. It was open of course. It's abt 3 miles from the highway (620). Then it's about another half mile to the point on the road adjacent to Whistler. GPS coords are in the file I sent you the other day; pls advise if you need me to resend.

Although the road appears to be in the activation zone, it's actually a fair bit below the ridge, so that estimation may be incorrect. Anyway we (Ansel tagged along and took most of the photos) climbed up to the top. The climb is all bushwhacking, first thru a laurel thicket, then clambering up a layer of scree. Fairly challenging but not quite as bad as Pyramid. As with Pyramid, there may be a better way – we just walked up and down the road a bit looking for what appeared to be the least unclimbable section.

At the summit, there’s plenty of relatively flat space. There are trees but they’re not very tall. Pix of scree and summit are attached. BTW I reduced all the pix but can send you the full res versions if you’re interested.

Got a few contacts on the way up the hill with the V8 as SOTA was pretty active today. When we got situated, I started with the RS 2m mobile attached to the hanging j-pole. That worked fine, got a few more contacts with it, but then it started to rain (wx forecast LIED). So I didn’t bother setting for HF and just packed up.

We then got back to Lucille and proceeded to Smoke Bluff. Easy enough to find – there’s a fork in the road shortly before it; the right side takes you toward the summit. Then there’s a closed green gate, at which point a left turn takes you up to the top. Overall it’s about 2 more miles from Whistler. GPS coords for the fork are in the file I sent you.

Smoke Bluff is pretty broad and flat, with no trees in the middle. There are some around the perimeter, but they’d be a little inconvenient to get at. Pix attached. There are some handy seats – foundations from the old fire tower. I just did 2m FM with the V8 and floppy antenna as it was still raining. While Ansel finished his sandwich in Lucille, I worked the 10m contest a bit and made 2 more contacts, one in Arizona and one in Venezuela.

…and it’s a darned good thing I did! I failed to “do the math” earlier – I figured, 7 contacts at Whistler and 6 at Smoke Bluff, why, that easily exceeds the 10 required for POTA. Except I neglected to consider that most of the latter activation were dupes of the first, so they only added up to 8 uniques. I activated the park by the skin of my teeth.

I won’t be running 10m mobile for a while. On the way down the hill, my antenna mount snapped. Antenna and feedline are fine but I have to fab up a new mount. And next time remember to remove the whip when I’m bouncing and jouncing on the 4wd roads. And dangit, the Ace hardware that replaced Rice does not carry 1/8” Al stock, and my supply is depleted.

Just north of the WMA we stopped at the Riddell Fire Tower. It’s open to the public to climb (at your own risk, it disclaims). So of course we did. Highly recommended. Pix attached.

CONCLUSION

A good set of activations, and a fine day in upstairs Virginia, somewhat dampened by the wx. We should definitely add these two to our annual rotation, but make them part of a larger combo as I mentioned above.

AFTERWORD

One of the SOTA pals asked if I was doing a solo trip this time and I said no, I had a muggle with me. Ansel found this humorous and countered, “I’m your emotional support muggle.” Perhaps that’s a better sobriquet than “dirty old man.”