The Epistemology of the Question Universe
Photo credit: Wikimedia Commons
When you watch the movie Hitch, you take it as a given that no one in that universe knows who Will Smith is. If you are going to question that, don’t see the movie.
You face a similar problem if you try to answer the following question:
You given a fair coin to flip one million times and get one million heads. What is the probability of your having gotten that result?
If this were a class and you’d like to get credit, you should say that there are two possible results for the first flip, two for the second, … and two for the millionth. This means there are 2^1,000,000 equally probable ways to flip a coin one million times. You only got one of those ways, so the probability is quite small: 1 / 2^1,000,000.
But what if your goal is to get the right answer rather than to get points on an exam?
Then, the problem itself needs to be scrutinized. First, note that every statement said by anyone ever is said by a fallible human and interpreted a fallible human.
So, we could (without adding or removing information) restate the problem:
A fallible human gives you what they say is a fair coin…
That phrasing doesn’t change any facts since the original problem statement didn’t say that G-d told you the coin was fair. But with this latter structure, we now realize that the fallible human probably fell hard.
We really ought not stop there. We are told that the coin was flipped one million times. Did they actually count? If flipping a coin and recording its results takes four seconds, then that’s 6.3 months of coin flipping 8 hours a day with weekends off. Would someone actually be able to do that? Or is it more likely for someone to feel like they must have flipped the coin for what felt like a million times to them?
You might object that the problem says that *you* flipped the coin. But presumably the question wasn’t written specifically for you. If you actually had flipped the coin yourself, you wouldn’t be reading a hypothetical question about it. The question is more likely directed at the full set of people who might be asked to answer it, not to you in particular. Unless you actually do remember flipping a coin for six months.
On the other hand, even if the coin was only flipped ten times, 1 / 2^10 is still a really small number: 0.000976562.
After getting the first few heads, it was probably worth questioning whether the coin is truly fair.
After ten heads, even a weighted coin becomes an unlikely possibility. A coin is unlikely to give ten heads in a row even if it is severely weighted. It’s now worth considering possibilities like the Super Bowl XVII coin:
[Head referee Jerry] Markbreit botched the coin toss during Super Bowl XVII. Dolphins captain Bob Kuechenberg called “tails,” and the coin came down “tails.” However, Markbreit became confused by the similar design of both sides of the coin: one side had two helmets and the other side showed two players holding helmets. Thus, he incorrectly thought “heads” had landed.
I challenge anyone to get ten heads in a row without a two-headed coin. In fact, this is the most likely correct solution to the original problem. A fallible human might in fact describe a two-headed coin as “fair” as long as both sides had an equal probability of showing. The right answer is 1, not 1 / 2^1,000,000. Our answer was off by about 100%.
Reporting the answer as 1 / 2^1,000,000 requires a little bit of statistics knowledge hiding in a giant vat of gullibility. What it says is that If an authority say that the coin is fair, no amount of evidence to the contrary will convince you otherwise. When reality collides with authority, you choose authority.
This has, I think, something to do with the barometer question:
“Describe how to determine the height of a skyscraper with a barometer.”
One student replied:
“You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building.”
This highly original answer so incensed the examiner that the student was failed immediately. The student appealed on the grounds that his answer was indisputably correct, and the university appointed an independent arbiter to decide the case.
The arbiter judged that the answer was indeed correct, but did not display any noticeable knowledge of physics. To resolve the problem it was decided to call the student in and allow him six minutes in which to provide a verbal answer that showed at least a minimal familiarity with the basic principles of physics.
For five minutes the student sat in silence, forehead creased in thought. The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn’t make up his mind which to use. On being advised to hurry up the student replied as follows:
“Firstly, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula H = 0.5g x t squared. But bad luck on the barometer.”
“Or if the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper’s shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper.”
“But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force T =2 pi sqr root (l /g).”
“Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up.”
Unfortunately, I don’t think this ever actually happened, but every smart student has had some intentional or unintentional version of this experience at some time.
Photo credit: Columbia Pictures
The student in this anecdote makes the mistake of taking the question honestly. The building is a lie, as is the professor’s interest in its height. The student is Eva Mendes going off script and asking “Why are you pretending to not be Will Smith? You don’t even have a fake mustache.”
Conceits are necessary if you want to enjoy a movie or ask certain questions for instructional purposes. I do wonder, however, whether inflexible assessment questions have the effect of indoctrinating students to be unable to see through false conceits in real life. People seem to have difficulty distinguishing actors from the people they portray. Is it so hard to believe that a grown-up student would have a hard time distinguishing between information reported by a computer or provided by a boss and Truth? Will they be able to question the question?
I’ve often thought about how fictional worlds in movies and television shows necessarily exist without the actors in question existing. There is no Will Smith in the “Hitch” universe. Otherwise, everyone would go, “HOLY SHIT, IT’S WILL SMITH!” instead of saying, “Hey, Date Doctor… help me out.”
I think it was Ocean’s 12 that played with this, wherein Julia Roberts’ character was recognized as looking similar to Julia Roberts, a “fact” the crew attempted to exploit as part of an escape plan. Part of the joke was that some doubted whether she could pull it off… whether she looked “enough” like Julia Roberts. I found that bit particularly clever.Report
It’s all the more problematic in Ocean X-type movies where part of the point is to go places without anyone noticing.
Earlier this summer, I saw a guy walking downtown who looked and dressed almost exactly like the Headmaster of Chilton High School from the Gilmore Girls.Report
But is it necessarily problematic? Movies are works of fictions. They often have feature time traveling, super heroes, and talking animals. Why is it problematic to believe that George Clooney movies all take place in universes where George Clooney, as we know him in our universe, does not exist? Movies wouldn’t exist without an ability to accept this.Report
I’d say it’s only problematic when it interferes with your enjoyment of the movie. You will probably enjoy the work more if you are able to give in to the assumptions provided by the alternate universe.Report
Is it problematic for you?
We all probably have certain pet peaves with regards to what we can suspend disbelief for and what we can’t. I struggle with fantasy because too often the rules feel inconsistent and that bugs me. But I have no real issue accepting the pretense that movie stars don’t exist within their own movies’ universes.Report
Occasionally it bothers me. The Bond movies are probably the worst offenders for me because even if they hadn’t cast a famous actor, the guy makes every attempt to attract attention.Report
Tristram Shandy: A Cock and Bull Story is quite a fun movie, if you haven’t seen it.
It’s set on the set of a movie of which we see little bits here and there. It’s got Gillian Anderson as Gillian Anderson, Steve Coogan as Steve Coogan, etc.Report
It also brings to mind the opening credit sequence of the A-Team where Dirk Benedict double-takes when crossing paths with a Cylon.Report
This reminds me of the muffin joke:
So, two muffins are baking in an oven. One muffin turns to the other and says, “Man, it’s hot in here!” The other says, “Holy shit, a talking muffin!”Report
@kenb
I love that joke. And when slightly edited, kids do, too.Report
TvTropes calls this the Celebrity Paradox.Report
Regarding the barometer problem, you left out the most clever answer: I would go to the building superintendent and offer him a brand-new barometer if he will tell me the height of the building!
I have to disagree with you on the coin-flipping problem though. As you almost correctly stated, the probability of flipping 10 consecutive heads with a fair coin is 0.0009765625. Expressed as a fraction, that is 1/1024. In contrast, the probability of winning the Pick 4 lotto is 1/10000, almost an order of magnitude more difficult, and that is won every time.
Do you know 710 people? If you do, and you asked them all to flip a fair coin 10 times, there is a 50/50 chance that one of them will flip 10 heads.Report
The ten flips were issued as a challenge. You’re absolutely right that it is doable. I just think that it’s unlikely that anyone reading this will have the patience to flip a coin a sufficient number of times to get ten heads in a row. It’s “hard” inasmuch as I don’t think anyone reading this will have that level of patience.Report
One in a thousand is unlikely, but hardly impossible. I’ve seen snake eyes come up twice in a row and that’s even longer odds than flipping 10 heads in a row. Unlikely things happen all the time because there are a lot of things happening all of the time. This is also known as the Wyatt Earp Effect.Report
Describe how to determine the height of a skyscraper with a barometer.
Rephrased to be less ambiguous.
Using the information the mechanism of the barometer provides, describe how to determine the height of a skyscraper with a barometer.Report
“Determine whether it’s going to rain. If it is, look up height on internet. If not, go to a coffee shop and look up height on internet.”Report
Alsotoo, the luckless undergraduate student has indisputable empiric proof that hell is exothermic and thus earns extra credit on his thermodynamics final as a consolation for his celibacy.Report
Look, if this post doesn’t start a slapfight over the Monty Hall problem, I will withdraw my ringing endorsement. 🙂Report
Always switch, damn it.Report
last time I had to prove that I just threw up my hands, coded an example, and ran it a few million times and printed out the results.
When he continued to argue I just said “Find the error in my code and we’ll talk”. 🙂Report
@morat20
Oh good, we are on the same side, like all right-thinking Americans.Report
I’d rather fight than switch!Report
@scarletnumbers
And all right-thinking non-Americans.Report
You got to convince people that Monty’s pick of the curtain to reveal is *not* random, then everyone should make sense.Report
That’s inherent in the problem definition. He *never* reveals the car, ergo his pick is not random — he deliberately chooses one of the non-car doors.
Even when you spell that out, a good chunk of the world jumps to 50/50, because “two doors, duh” and shan’t be moved by math or simulations or bricks to the forehead.Report
The problem is that the person describing the Monty Hall problem often fails to adequately emphasize that fact; they usually just say “before you pick, he opens one of the doors”, implying that he opens a door at random.
And if the person you’re describing the problem to still has trouble, say “okay, let’s suppose there’s a hundred doors. You pick one, then Monty opens ninety-eight of the others, leaving only two: the one you picked and one other one. Now, Monty knows which door has the car, so why do you think he left that one closed?”Report
“The problem is that the person describing the Monty Hall problem often fails to adequately emphasize that fact; they usually just say “before you pick, he opens one of the doors”, implying that he opens a door at random.”
This, but I would put is “…often fails to state the condition…” Without this statement, there is no reason why the condition must be there. I simply don’t buy it that the condition is implied. This is an ex post facto argument slapping a Band-Aid on a gaping hole. Suppose the “right” answer required that Monty not act the same way every time. If anyone defended the “wrong” answer on the grounds that it is implied that Monty acts the same way every time, people would be mocking this person for coming up with a lame excuse for his wrong answer.
Note also that the classic Marilyn Vos Savant lacks the necessary statement. It is actually fairly typical of her to pose a poorly worded question, followed by an authoritatively worded answer.Report
Actually, even if he opens a door at random you’re still better off switching, as long as he doesn’t show the car.
You pick one of three (33% chance you get the car), which means there’s a 2/3rds chance the prize is behind one of the other two doors.
Since the prizes aren’t shuffled, those odds never change.
By opening a door, Monty has show you that of the doors with a 66% chance of holding the prize, one does not. Ergo the remaining door will have the car 66% of the time.
The ‘trick’ is both realizing the initial odds never change and grasping that by informing you of what is behind one of the remaining doors, you effectively get to choose TWO doors by switching.
But anyways, the game wouldn’t make a lick of sense if Monty opened the door to show the actual prize. “Pick a door. I’ll open a door, and then you can keep your door or switch to the other unopened door. And behind my door — a CAR! So do you want to switch? Wait, why did you just flip me off?”Report
morat20:
Uh oh, someone was wrong on the Internet three days ago!
If Monty’s choice is random, then it makes no difference whether you switch. Think of it this way: since the job of opening the door now requires no knowledge, anyone in the world could do it, such as another contestant — or you!
So imagine the show goes like this: You pick a door to be the “first” door. Then you pick another door to be the “third” door. Then you open the remaining door (the “second” door), revealing nothing.
Which door has a better chance of concealing the prize — the “first” one or the “third” one? I hope it’s clear that the difference between them is arbitrary. The assignment of the three doors could have happened in any order, or simultaneously. All that really happened is that a door was shown to be vacant, leaving two equivalent doors left.
If there were a million-dollar raffle in which one of a thousand tickets is the winner, is it better to pick earlier or later? What if you picked the very first ticket sold, someone else got the very last one, and after that all the ones in the middle were shown to be duds? Would you pay the last person in line for their ticket, and if so, how much? A thousand dollars?
By contrast, the classic Monty Hall is equivalent to someone who knows the winner being given all the tickets except yours. This person is then told to throw out all but one ticket, after which (to his surprise) you are given the chance to switch tickets with him.
Knowledge makes all the difference — which I suppose brings things to the original topic in a way.Report
That was so two years ago.Report
Cute. Some comments.
Is it reasonable to ask whether you have a fair coin at some point? Of course, that’s what the low probability of your outcome is telling you. The point is even stronger if this is the only time that experiment has been done.
However, in a universe where, say, coin-flipping experiments have been ongoing for millennia, then at some point you’re going to end up with at least several very unlikely outcomes. Take your 10 flips challenge—given maybe a few thousand experiments, one is bound to hit. And for the one that does hit, there’s probably no real reason it couldn’t have happened on the first try.
So you have to be careful in how you interpret unlikely events—unlikely does not mean impossible. And jumping to that conclusion can lead to weak causal hypotheses: God did it, the stars are aligned a certain way, I didn’t wash my underwear for a year, I should have sacrificed a virgin, etc.
Understanding scientific laws in the abstract pure sense is immensely useful, even if you can’t achieve that purity in everyday real-life.
As for the student who answered about the barometer. I mean, c’mon. We all know he’s being a smart-ass and that the teacher is trying to get him to understand the physical laws of air pressure. The question even tries to connect the concept to a real situation, which is a step above the loads of typical physics questions that don’t even attempt to make you think.
And just because the student can rattle of several alternative methods for getting the height of the building, he still hasn’t hit on the concept of air pressure. Following your own advice from the post, I would conclude it’s unlikely to be just chance that he knows so many ways to solve the problem except for the one the teacher is looking for. He’s dodging. Because he is obviously smart but was too lazy to study the material and is looking for his cleverness to save him. (Either that or he’s just antagonizing the teacher.)
So what are you suggesting should be done? Should every possible ambiguity be removed from all questions so that there is no wiggle room? (Are you a lawyer, by chance?) Should there be follow-up questions that ask the student to make judgements based on the data?
Or should it go the other way, and should questions remain open, but if students can find a way around it, they may be rewarded? (Again, are you a lawyer?)
I’m just not sure what you’re getting at. But it was a fun read.Report
I believe Mr Bath is a college professor.Report
I guess that would make him Dr Bath.Report
I was but am no longer a professor. My wife is still a professor.Report
I’m not saying ten heads isn’t possible. I’m saying that it is a clue that you might not be dealing with an ordinary coin.
Alternatively, if someone comes to you and says “I flipped a coin ten times and got ten heads”, I’d be suspicious about the person’s honesty or the coin’s nature. These suspicions require you to be willing to call bullshit on people though, which isn’t something we teach, I think.
Something might be possible, but sufficiently unlikely that it’s worth thinking of alternative explanations beyond what you’re used to considering with a certain type of problem.
>barometer problem
My beg is that he was antagonizing the teacher rather than not having studied. At the same time, I empathize with the student for feeling that he is being told “answer this question in the way I want you to rather than to get the answer” because that is what is happening. If instructors asked students questions only when they were genuinely interested in an answer, there would be no tests.
So what are you suggesting should be done? Should every possible ambiguity be removed from all questions so that there is no wiggle room?
If it were me, a student who gave me the rope answer would get full credit. The next year, I’d change the question to something like “How could you use the barometric pressure at the top and bottom of a building to calculate the height of a building?”
—
I’m just not sure what you’re getting at. But it was a fun read.
That’s enough for me! As to what I’m getting at, let me try an example. I taught in the business school where we sometimes did case studies. These are written up to include a bunch of information from various sources. They typically have a lot of ambiguity, and there are a lot of directions students can go. One thing I noticed and could never get students to overcome was that when the protagonist of the case said something, the students almost always took that as gospel. It was hard to get them to look at other information in the case that might have contradicted that. I think they took certain bits of information most directly provided as given, set-in-stone assumptions. I never figured out a way to make them less willing to trust the views provided by protagonists, and I think this the way we ask questions in other settings might be a reason.Report
“If instructors asked students questions only when they were genuinely interested in an answer, there would be no tests.”
On the contrary, those make the best tests.Report
So about the coin-flipping, I think you’d suggest that there should be a follow-up question, like, “Given the probability you calculated, do you think you were actually given a fair coin?” And I’d agree with that!
For the barometer question, if the teacher is trying to measure the student’s gains in atmospheric physics, then he really can’t reward the kid for answers that don’t address that, can he? If the class is actually Fluids and Dynamics, and the student can find end-runs around all the important questions, would an A or B in that class be representative of what the student truly understands of that subject?
On the other hand, what if the course is Thinking with a Physics Mindset? Then the student’s answers would be appropriate. But in that case, the teacher also wouldn’t be pushing for a specific answer.
I guess what I’m saying is that if you’re conducting a course with the goals of communicating specific content knowledge and assessing how much of that knowledge students absorb, then that’s not a very good environment for encouraging outside-the-box thinking.
In the business class you described, students were likely just trying to get to the “correct” conclusions in order to get a good grade. Getting them to question the information from the protagonists and think through the ramifications would require giving them the space to get things wrong with minimal or no punishment. However, that would then make assessing them that much more difficult, at least within the timeframe you likely had for the course. (Of course, I know nothing of the class other than what you mentioned, so I’m completely spit-balling here. And I probably got it all wrong!)
So I empathize with your lament about the general inability of students to question assumptions and weigh competing possibilities. I just think that the problem is much deeper than simply adjusting assessment questions. (Though it wouldn’t hurt!)Report
if you’re conducting a course with the goals of communicating specific content knowledge and assessing how much of that knowledge students absorb, then that’s not a very good environment for encouraging outside-the-box thinking.
I agree! What I was trying to get at though is that these seem to be the only form of questions students encounter until someday they enter a more realistic setting and then miss other solutions or trust as given information that isn’t perfect.
In the business class you described, students were likely just trying to get to the “correct” conclusions in order to get a good grade.
I personally don’t think this is likely in the situations I have in my head right now (though perhaps it applies to others). Cases are typically written with a strong narrative, and it’s natural to just go along with the story even if you don’t care about your grade. In ungraded discussions, they seem to have the same issues.
So I empathize with your lament about the general inability of students to question assumptions and weigh competing possibilities. I just think that the problem is much deeper than simply adjusting assessment questions. (Though it wouldn’t hurt!)
Yes, I don’t present much of a solution in the post. I only articulated what I see as a problem. I am always hopeful the comments will expose some direction in which to go.Report
My high school science teacher told us a story about questioning assumptions and honestly reporting data, with a little bit of time management thrown in:
In one of her college lab-based courses, the students had to evaluate a series of solutions for a property (let’s say pH; I don’t really remember). Each of the solutions was labeled and even pre-google, it wouldn’t be that hard to determine what the pH *should* be. When my teacher was writing up her results, the night before it was due, of course, one of her results was wrong- the pH was significantly different than it should have been. Now, had there been time, she would have gone back and re-tested the sample. But there wasn’t, and she *knew* what it was supposed to be. So she did what a lot of people do: she fudged the numbers a bit. When she got back the lab report, she had a big fat zero and the word “LIAR” written on it. (The professor had intentionally mis-labeled the sample) It was an expensive lesson, but she learned:
1) Report your results, whatever they are
2) Question your “givens”, especially if the data seem to contradict them
3) Don’t procrastinateReport
I would conclude it’s unlikely to be just chance that he knows so many ways to solve the problem except for the one the teacher is looking for.
That is, it’s unlikely to be just chance that he’s saying all the ways to solve the problem except for the one the teacher is looking for. He knows what answer the teacher wants, but he refuses to say it. That’s called being a smartass.Report
While we’re being
pedanticdetailed, isn’t the answer “1” for a different reason? You have flipped a coin 1M times and gotten 1M heads. It happened. The probability that it happened is 1.Now the probability that it will happen again is another story, of course!Report
That occurred to me too, and I tried to revise the verb tenses to avoid that issue by asking what the probability of having gotten such an unlikely result (before you actually got the result).Report
The wrestling concept of kayfabe covers, I think, what you going for with the Will Smith example. And which is, imo, not at all like the skepticism one should hold the claim of encountering a fair flipped coin result of a million fair flipped coin, which is yet still different from an apocryphal story of some smart-aleck student.Report
Well…
2^1,000,000.
Let’s say it takes 2 seconds to flip a coin (flip, catch, slap reveal). That’s probably generous.
There are 31,536,000,000,000 seconds in a million years. There are 3.1536 x 10^19 seconds in a trillion years.
2^ 1,000,000 is sorta a whole bunches of shitloads bigger than 3.1536 x 10^19. By a mindblowingly, astronomically hugemongus amount. This isn’t peas vs. a battleship, this isn’t peas vs. the Sun. You can’t even come up with an imaginable comparison.
The Universe, actually, has only been around for 13.798 billion years or so.
The law of large numbers pretty much dictates that the probability of flipping a million heads in a row will never occur before the heat death of the universe. There’s just literally not enough time to have enough total trials to have one possible one of those trials come up with that many heads in a row.
The Universe doesn’t have an infinite life span.
Put another way, let’s take the low figure for the heat death of the Universe at a trillion years. At that time scale, the current existing 13 billionish years is almost a rounding error. If we could have started at the beginning of the Universe and flipped a coin every 2 seconds until the end of everything, we’d get a total of
157,680,000,000,000,000,000,000 trials
In that many trials, you’d expect your longest run of either heads or tails to be about a gadzookizillion times smaller than 2^1,000,000. More here, money quote:
“I played around with the numbers a bit and found that if you flipped a coin 2,000,000,000,000,000,000,000,000,000, or 2 x 10^27, times, you should expect your longest run of heads to be around 90.”
To get to a million heads in a row, well, I’m not sure the combox can fit that many zeros.
Given the time scale of the Universe, if somebody handed you a coin and you flipped it a million times and got a million heads, you could state with certainty your ability to perceive the Universe was being fucked with, or you just witnessed an honest-to-God miracle.
Even a fixed coin would not come up heads a million times in a row.Report
I wonder how many times you could flip a newly minted coin before it would deform or break into something that couldn’t realistically be flipped anymore.Report
GAO sez the expected lifespan of a dollar coin is about 30 years.
Let’s say that’s when it’s considered worn enough to not be optimal for currency any more, and that you could I dunno…quintuple? that time before it was worn enough that you couldn’t distinguish heads from tails any more.
That would give you 150 years, which at the aforementioned 2 seconds per flip gives you 2,365,200,000 flips.
That’s not going to give you a very long run 🙂Report
Which of these coin-flip sequences is least likely to occur?
1) HTHTHTHT
2) HHHHHHHH
3) HTTHHTHT
It’s a trick question. All have the same probability of occurrence.Report
If it’s a fair coin, of course. If there’s (say) a 10% chance it’s a 2-headed coin, 2 is the most likely, a bit more than 10%.Report
On the other hand, there’s Benford’s Law.Report
I’m reminded of the following story/evaluation (attribution would be appreciated):
A coin is flipped ten times; each time heads appears (see note 1, below, after you read the evaluation). You must bet on the next coin flip. How do you bet?
The Evaluation:
The Optomist says, “Heads! You can’t fight the trend/luck/hot streak!”
The Pessimist says, “Tails! This can’t go on forever!” (see note 2)
The Realist says, “It doesn’t matter, there’s an equal chance. The Optomist and Pessimist are fools!”
The Gambler says, “The Realist is correct–it’s 50-50. However, I’m choosing heads…there may be something wrong with that coin or the flipping method…”
(1) I’ve actually witnessed something like this. I was sitting at a blackjack table in Biloxi, right next to a roulette table. Red came up eleven times in a row (and that’s less than a 50-50 chance, due to the 0 and 00 on the roulette wheel).
(2) At that roulette table, litterally no one was making money. They were all Pessimists…Report
Of course no one was making money. They were playing roulette.Report
Last time I played roulette, red 23 hit three times in ten rolls.
Luckily my “system” consists of betting on red 19, 21, 23, 25, 27, and on the outside, red, odds, and 19-36. I got paid handsomely in four places each time it hit, walked away $200 richer, and the buffet was on me that day.Report
This is an excellent formulation, Bill.
I think the Gambler’s take best represents my views.Report
I sometimes wonder if Lt. Dennis Mello ever suggested to Sgt. Jay Landsman that he should join a gym.Report
Dr. Bath, this was an excellent read. Please keep up the good work, I’ve enjoyed all your post thoroughly since I returned to these haunts.
BTW, have you read the book, “Against the Gods”? I think you’d really enjoy it.Report
I haven’t. I think I remember it coming out. I can’t promise I’ll get to it, but I’ll add it to my list of things o get to. Thanks for the recommendation.Report