Monty Hall Paradox explained

This is a well known problem based on a 1960s US television game show called Let’s Make a Deal. How closely it resembles that particular show, I don’t know, but it’s not relevant, because it’s easy to imagine. The show’s host’s stage-name was Monty Hall, hence the name of the puzzle.

 

In 1975, an American statistician and professor at the University of California, Berkeley, Steve Selvin, published a short article on the Monty Hall Paradox in The American Statistician, which he saw as a curiosity for a very select group who would appreciate its quirkiness and counter-intuitive answer. He received some criticism, which he easily countered.

 

Another totally unrelated (weekly) periodical, Parade magazine, with a circulation in the tens of millions, had a column called Ask Marilyn, who specialised in solving mathematical puzzles, brain teasers and logical conundrums sent to her by readers. She was Marilyn vos Savant, and entered the Guinness Book of Records in the 1980s as the woman with the highest recorded IQ (185). I obtained all this information from Jim Al-Khalili’s book, Paradox; The Nine Greatest Enigmas in Physics.

 

Someone sent Marilyn the Monty Hall puzzle and she came up with the same counter-intuitive answer as Selvin, but she created an uproar and was ridiculed by mathematicians and academics across the country. Al-Khalili publishes a sample of the responses, at least one of which borders on misogynistic. Notwithstanding, she gave a more comprehensive exposition in a later issue of Parade, emphasising a couple of points I’ll come to later.

 

Now, when I first came across this puzzle, I, like many others, couldn’t understand how she could possibly be right. Let me explain.

 

Imagine a game show where there is just a contestant and the host, and there are 3 doors. Behind one of the doors is the key to a brand-new car, and behind the other 2 doors are goats (pictures of goats). The host asks the contestant to select a door. After they’ve made their selection, the host opens one of the other 2 doors revealing a goat. Then he makes an offer to the contestant, saying they can change their mind and choose the other door if they wish. In the original scenario, the host offers the contestant money to change their mind, upping the stakes.

 

Now, if you were a contestant, you might think the host is trying to trick you out of winning the car (assuming the host knows where the car is). But, since you don’t know where the car is, you now have a 1 in 2 chance of winning the car, whereas before you had a 1 in 3 chance. So changing doors won’t make any difference to your odds.

 

But both Selvin and vos Savant argued that if you change doors you double your chances. How can that be?

 

I found a solution on the internet by the Institute of Mathematics, giving a detail history and a solution using Bayes’ Theorem, which is difficult to follow if you’re not familiar with it. The post also provides an exposition listing 5 assumptions. In common with Al-Khalili, the author (Clive Rix from the University of Leicester), shows how the problem is similar to one posed by Martin Gardner, who had a regular column in Scientific American, involving 3 prisoners, one of whom would be pardoned. I won’t go into it, but you can look it up, if you’re interested, by following the link I provided.

 

What’s important is that there are 2 assumptions that change everything. And I didn’t appreciate this until I read Al-Khalili’s account. Nevertheless, I found it necessary to come up with my own solution.

 

The 2 key assumptions are that the host knows which door hides the car, and the host never picks the car.

 

So I will describe 3 scenarios:

 

1)    The assumptions don’t apply.

2)    We apply assumption No1.

3)    We apply assumptions 1 & 2.

 

In all 3 scenarios, the contestant chooses first.

 

In scenario 1: the contestant has a 1 in 3 chance of selecting the car. If the contest is run a number of times (say, 100 or so), the contestant will choose the car 1/3 of the times, and the host will choose the car 1/3 of the time, and 1/3 of the time it’s not chosen by either of them.

 

Scenario 2: the host knows where the car is, but he lets the contestant choose first. In 1/3 of cases the contestant chooses the car, but now in 2/3 of cases, the host can choose the car.

 

Scenario 3: the host knows where the car is and never chooses the car. Again, the contestant chooses first and has a 1 in 3 chance of winning. But the host knows where the car is, and in 1/3 of cases it's like scenario 1. However, in 2/3 of cases he chooses the door which doesn’t have the car, so the car must be behind the other door. Therefore, if the contestant changes doors they double their chances from 1 in 3 to 2 in 3.

Will the 21st Century be a turning point in human history?

 The short answer, I believe, is Yes, but whether it will be positive or negative is up for conjecture. If history is any guide, I’d have to say things don’t look particularly promising. There have been a number of things I’ve read recently, and viewed on TV, from various sources that have made me reflect on this, and it’s hard to know where to start. 

Maybe I’ll start with something I wrote on Facebook recently, which was the seed for this rumination.

 

Humanity has always had both the capacity and inclination for self-destruction. It is our Achilles heel. One can't help but think that the 21st Century is our turning point, one way or the other.

 

There are lots of examples, the Roman Empire being one of the most cited, but also the ancient Egyptians and the Mayans, not to mention Easter Island. Curiously, I’ve just started watching Foundation, on Apple TV, based on Isaac Asimov’s famous books, which is premised on the fall of a future galactic empire founded and run by humanity.

 

But there is another TV series by the BBC called Capital, very contemporary, which I’ve also just started watching, and seems to encapsulate our current situation. I’ve only watched one episode, which centres on a single street in England, but is rendered as a microcosm of global politics and social dilemmas. 

There is the corporate middle manager whose ambition and greed is only outdone by his wife, who mentally spends his money before he’s even earned it. There is the refugee from Zimbabwe who is working illegally, therefore exploited by an ‘agent’, while she faces imminent deportation even though she fears death on arrival. Something that refugees in Australia can readily identify with. There is the Pakistani corner shopkeeper, who is a diligent neighbour, with 2 sons, one who has become religiously conservative and the other who has started, but not completed, 3 university degrees (I can identify with that). He’s the target of a stalker, covertly photographing him and his family. Like everyone else in the street, he’s receiving postcards with the ominous warning, We Want What You Have. In other words, there is an undercurrent of class envy which could fester into something more sinister. Another of the recipients is an elderly woman, whose son and daughter have all but abandoned her, and who is facing terminal illness, but she’s inherited the sin of living in a capitally inflated home.

 

Also, on TV recently, I watched a programme on (Australia’s) ABC 4 Corners, called The Pandora Papers, which is about tax havens for the ultra wealthy and powerful, and really identified an ‘alternative universe’, as one commentator described it, that the rest of us are largely unaware of. The programme showed how, in Australia, unidentified foreign investors are driving the price of homes beyond the reach of ordinary citizens who live here. There have been other programmes about corruption in the food industry in Europe, which goes beyond the borders of Europe.

 

I’ve read other stories in newspapers, and what they all have in common is inequality. Curiously, Philosophy Now (Issue 145 Aug/Sep 2021) had as its theme, existentialism, but included an article called The Stoic’s Lacuna by Alex Richardson, a History teacher at Croydon, UK. Its relevance to this topic was a reference to the Greek stoic, Epictetus, who said, “Seek not that the things which happen should happen as you wish but wish the things which happen to be as they are, and you will have a tranquil flow of life.” In other words, accept one’s lot in life and stop whinging.

 

Richardson’s essay extends into the modern day by referencing Katherine Birbalsingh (given the sobriquet, Britain’s strictest headmistress), Dr Michael Sugrue and Jordan Peterson as ‘modern day stoics’, who all advocate in varying degrees, that inequality is the natural order of things. Birbalsingh may be the most liberal of them, when she says, “Of course the world is run by an old boys’ network, and of course it’s not fair.” I admit I know nothing about her outside Richardson’s essay, but he puts her in the same sentence, therefore category, as former Navy SEAL, Jocko Willink, who effectively argues that a person’s day-to-day struggle with paying off a mortgage and generally making ends meet is completely disconnected from ‘political management of the economy’. 

 

Peterson is someone I’m more familiar with, who effectively argues that inequality is an evolutionary consequence of the survival of the fittest, not only in the natural world but in human affairs. People, especially males, get to the top of the heap, where they are especially attractive to females, who copulate and subsequently procreate with them to ensure the survival of both parties’ genes. As it happens, this exact scenario is played out by one of the families represented in the aforementioned TV show, Capital.

 

Richardson believes that Peterson is a ‘follower’ of the ‘Pareto Principle’, expressed in the Bible (both Mark and Matthew): “For to everyone who has will more be given, and he will have abundance, but from him who has not, even what he has will be taken away.” In other words, the rich get richer and the poor get poorer. Or as Richardson puts it, “wealth and power naturally accumulate in the hands of a few exceptional individuals at the top” (my emphasis).

 

But, according to Richardson, Dr Michael Sugrue is the most blunt and dispassionate, when he said in a popular lecture on Aurelius that ‘Stoicism teaches us that the social structure is “not our problem” and that, “if God, or nature, or whatever is controlling the world makes you a slave then be a good slave.”'

 

The common thread in all these admonitions, is that they are made by people who see themselves among the privileged elite, who would never contemplate that what they advocate for others could befall them.

 

I think inequality drives injustice, corruption and an upside down economy. To give an example, Italy. It’s well known that there is both social and wealth disparity between the north of Italy, which is the capital of supercars and high fashion, and the south of Italy, which is the home of agriculture and the country’s food bowl. But this dichotomy is worldwide. The production of food, which is essential, is one of the lowest paid occupations in the world.

 

Now, let’s add another factor, called climate change. I don’t find it altogether anomalous that climate change has a dichotomous effect on humanity. It’s the consequence of all the ‘progress’ we’ve made since the industrial revolution, and it’s a juggernaut that can’t be stopped. Yet it will affect the poorer nations first. As the Prime Minister of Samoa, Fiame Naomi Mata’afa, recently said, ‘Pacific Islanders don’t want to be the canary in the coalmine for climate change’. If one looks at Italy again, one could argue that supercars have contributed to climate change and the agriculture sector will bear the consequences.

 

I recently did an online course provided by New Scientist, called Greener Living, which was ostensibly about climate change, its causes and its effects on future generations. According to the people running the course, it will require enormous changes to the way we live, including what we eat.

 

In the 25-26 September issue of the Weekend Australian Magazine (a Murdoch publication, btw) there was an interview with 33 year old Anika Molesworth, a scientist who also runs a farm near Broken Hill in NSW. She says that modelling for 2050 (based on nothing changing) would see 30% decrease in rainfall and 2 months of days above 40C, which would make the property effectively inoperable. But she also claims we have the means at our disposal to change this outcome, and she’s a founding director of Farmers for Climate Action. She’s frustrated by the missed opportunities in our country for renewable energy; we have a government that is stubbornly resistant to changes to the status quo.

 

I made an allusion before to the well known meme of evolution as the survival of the fittest, but much of evolution has occurred through symbiosis. Your body is an entire ecosystem to organisms that thrive in order for you to live, largely without your cognisance. I know from a working lifetime in engineering that successful projects are the result of people collaborating and working together. Environments, including our political environments, where people are antagonistic and work against each other, achieve little except blame and finger-pointing. A perfect example of that is the current political climate in America.

 

If we don’t want to self-destruct, we need to work together, punish corruption that erodes the wealth and agency of ordinary people, adopt sustainable economic models, not dependent on infinite consumerism and keeping people in debt for their entire productive lives. If we stick to the mantra that inequality is the ‘natural order’, we will fail and it will ultimately be catastrophic, worse than the Roman Empire, the Egyptian empire or the Mayan empire, because it will be global.

Tips on writing sex scenes

 Yes, this is a bit tongue-in-cheek, or tongue in someone else’s cheek to borrow a well-worn witticism. This arose from reading an interview by Benjamin Law (Aussie writer) of Pulitzer-prize winning author, Viet Thanh Nguyen, who briefly discussed writing sex scenes. He gives this advice: 

Not being utterly male-centred, if you happen to be a man or masculine. Not being too vulgar. Don’t be too florid. And humour always helps.

 

Many years ago (over a decade) I attended a writers’ convention, where there are always a lot of panel discussions, and there was one on ‘How to write sex scenes’, which was appropriately entertaining and unsurprisingly well attended.

 

Even longer ago, when I attempted to write my first novel, with utterly no experience or tuition, just blindly diving in the deep end, there was the possibility of a sex scene and I chickened out. The entire opus was terrible, but over about 3 years and 3 drafts I taught myself to write. I sent it to a scrip-assessor, who was honest and unflattering. But one of the things I remember is that she criticised me for avoiding the sex scene. I was determined to fix that in subsequent attempts. I still have a hard copy of that manuscript, by the way, to remind myself of how badly I can write.

 

But there are a couple of things I remember from that particular panel discussion (including a husband and wife team on the panel). Someone asked for a definition of pornography, and someone answered: the difference between erotica and pornography is that one you don’t admit to reading (or watching, as is more often the case). So, it’s totally subjective.

 

The first editor (a woman) of ELVENE, took offense at the first sex scene. I promptly sent the manuscript to 2 women friends for second and third opinions. Anyway, I think that you’ll invariably offend someone, and the only sure way to avoid that is to have all the sex happen off the page. Some writers do that, and sometimes I do it myself. Why? I think it depends on where it sits in the story, and is it really necessary to describe every sexual encounter between 2 characters, who start doing it regularly?

 

The other thing I remember from that panel is someone explaining how important it was to describe it from one of the character’s points of view. If you describe it from the POV of a ‘third party’, you risk throwing the reader out of the story. I contend that the entire story should be told from a character’s POV, though you can change characters, even within the same scene. The obvious analogy is with dialogue. You rarely change POV in dialogue, though it’s not a hard and fast rule. In other words, the reader’s perspective invariably involves speaking and listening from just one POV, as if they were in the conversation. The POV could be male or female - it’s irrelevant - but it’s usually the protagonist. I take the same approach to sex scenes. It’s risky for a man to take a woman’s POV in a sex scene, but I’ve done it many times. 

 

I often take the POV of the ‘active’ partner, and the reader learns what the other partner is experiencing second-hand so to speak. It generally means that the orgasm is described from the other partner’s perspective which makes it a lot easier. If they come in unison, I make sure the other one comes fractionally first.

 

I don’t write overlong sex scenes, because they become boring. Mine are generally a page long, including all the carryon that happens beforehand, which is not intentional, just happenstance. I wrote a review of Cory Doctorow’s sci-fi novel, Walkaway, a novel (full title) which has a number of heavy sex scenes which I did find boring, but that probably says more about me than the author. I’m sure there are readers who find my sex scenes ‘heavy’ and possibly boring as well. 

 

I have some rules of my own. They are an interlude yet they should always serve the story. They tell us something about the characters and they invariably have consequences, which are often positive, but not necessarily so. There is always a psychological component and my approach is that you can’t separate the psychological from the physical. They change the character and they change the dynamic of a relationship. Some of my characters appear celibate, but you find them in real life too.

 

I take the approach that fiction is a combination of fantasy and reality and the mixture varies from genre to genre and even author to author. So, in this context, the physical is fantasy and the psychological is reality.

 

One should never say ‘never’, but I couldn’t imagine writing a rape scene or someone being tortured, though I’ve seen such scenes on Scan-noir TV. However, I’ve written scenes involving sexual exploitation, to both men and women, and, in those cases, they were central to the story.

 

Lastly, I often have to tell people that I’m not in the story. I don’t describe my personal sex-life, and I expect that goes for other writers too. 

There are 2 paradoxes in relativity

 This post is based on an interaction I had on Quora. When people discuss the twin paradox (like I did), they always discuss the difference in time, each twin experiences relative to the other; they don’t discuss the difference in distance they experience, which is a logical consequence of the same Lorenz transformation. 

However, someone on Quora conjectured that in reality, the clocks actually measure time at the exact same rate and it’s the distance that changes. Mathematically, there is nothing wrong with this argument, so why do I disagree with it?

 

The twin paradox, of course, is a thought experiment, but there is a real-life experiment where this happens, which I’ve also discussed in another post, and that is the half-life of muons in the Earth’s atmosphere. According to the scientists observing them on the Earth’s surface, there are too many of them, which means they took longer to decay (the half-life got longer). But, from the muon’s perspective, their half-life hasn’t changed, it’s just the distance to the Earth was shorter. Now, this creates a paradox, which is equally manifest in the twin paradox thought experiment. 

 

How can the distance change depending on whose perspective you take? I don’t believe it does. I contend that the moving observer ‘measures’ a shorter distance based on their clock slowing down. But, you’ll say, the whole point of relativity is that no one knows who is moving and who is stationary. Well, there is a reference point, which, in the case of the muons, is the gravitational field of Earth. A body in free fall experiences maximum ‘proper’ time and any deviation from that causes a clock to slow down. The fact is, because the muons are travelling at high fractional light speeds, their deviation from maximum proper time is greater than the observers on the ground. I believe this logic also applies to the travelling twin in the thought experiment.

 

If you take this to extremis, someone could hypothetically travel across the entire galaxy in their lifetime if they travelled fast enough (without exceeding the speed of light). From their perspective, the distance would shrink astronomically, but if they returned to Earth, eons would have passed in their absence. I should point out that science fiction writers (like myself) routinely ignore this fundamental consequence of relativity. We ‘imagine’ that humanity has discovered ‘new physics’ to overcome this ‘problem’

 

I’m not sure how heretical it is to argue this: that time changes but the distance doesn’t. if one goes back to the argument that started this rumination: when the twins reunite their respective clocks disagree and the space-faring twin is noticeably younger than their counterpart. However, they not only disagree on the elapsed time, since the space-faring twin’s departure, they also disagree on how far that particular twin travelled. BUT, now that the space-faring twin is back in the same frame of reference as their Earthbound twin, they concur on the distance, yet still disagree on the time elapsed, evidenced by their clocks and age difference.

 

 

Addendum: Ian Miller (PhD in Chemistry and a self-described 'theoretician' on Quora) argues that it's the ruler that changes relativistically and not the distance travelled. And I'd contend that the clock is effectively the ruler.

Existentialism in everyday life

This is a post I wrote on Quora, in answer to the question:

 

If I believe that life is completely meaningless, am I an existentialist or a nihilist?

 

My answer was ‘upvoted’ by Frederick Dolan, who is Professor at UC Berkeley, after I commented on his own answer to the same question.

 

 

Existentialism is often associated with the absurd, thanks to Camus mostly, which is a short step from nihilism in many people’s minds. There is a view among the scientific community that the Universe is meaningless and that the presence of sentient beings is just a freak accident. Paul Davies refers to this view as the ‘absurd universe’, and I tend to agree with him. 

 

The point is that we find meaning in living our lives through relationships and projects, and even through career or vocation. Viktor Frankl elaborated on this in his book, Man’s Search for Meaning, and added a third way, which is through dealing with adversity. This last ‘way’ might appear ‘absurd’ at first glance, but it’s the premise of virtually every story ever told; it’s a universal theme.

 

Sartre argued that we create our own ‘project’ just by living our lives, and this is what existentialism means to me. So I would contend that there is a direct connection between existentialism and the importance of finding meaning in our lives. In fact, having a ‘purpose’ is considered essential to mental health, according to psychologists. If the Universe has any meaning at all, it’s because it created sentient beings who find meaning against the odds that science tells us are astronomical, both literally and figuratively. Existentialism is about finding purpose in an absurd universe, which is the opposite of nihilism.

A philosophical exploration of Type A and Type B time

 This arose from a question referred to me on Quora. As part of my discussion, I wondered into philosophical territory originally posited over a century ago by a forgotten philosopher, J.M.E. McTaggart, thanks to A.C. Grayling (English philosophers seem to have a predilection for using initials). It seems to fit seamlessly into my own particular philosophy on the relationship between time and consciousness.

 

The original question on Quora, asked by Adriana Moraes (from Sao Paulo, Brazil):

 

How does the past, present, and future exist simultaneously?

 

I don’t believe they do. In fact, I contend that past, present and future are only meaningful concepts in some creature’s mind; which means that I don’t believe it’s a cognitive state unique to humanity.

 

We are only aware of the past because we have memories. In fact, without memory, you wouldn’t know you are conscious. Consciousness exists in a constant present, so time for us is always ‘now’. This, of course, applies to all sentient creatures. For all events that we witness or observe, ‘now’ is ephemeral – they become the past as soon as they happen - which is demonstrated every time someone takes a photo. We say it ‘freezes time’, when in fact, it records an event that would otherwise vanish.

 

Past, present and future require a reference point, and consciousness provides that reference point. We imagine futures, and curiously, the same part of the brain that imagines what might happen, conjures up memories of what has happened. This makes sense when one realises that we attempt to predict the future based on what we have experienced in the past. 

 

Raymond Tallis, who has a background in neuroscience and writes books as well as a regular column in Philosophy Now, makes the observation that our ability to ‘imagine’ future ‘possibilities’, and select one to make ‘actual’, is the very definition of free will, only he calls it 'agency'.

 

In 1908, an Oxford philosopher, J.M.E. McTaggart published a paper titled, On the Unreality of Time in the journal, Mind (ref: A.C. Grayling, The History of Philosophy, 2019). McTaggart argued that there are 2 types of time: Type A, which is based on using the ‘present’ as a reference point for ‘past-present-future’; and Type B, which is just the ordering of events into ‘earlier than/later than’. He contended, in effect, that because ‘now’ is constantly changing, you get contradictions with Type A and Type B (which is perceivably 'fixed'). I’ve over-simplified his argument for brevity, and given it my own interpretation, which is that you can’t have both Type A and Type B. However, I contend that Type B time is just Type A time without consciousness, which resolves that particular paradox.

 

Most physicists, if not all, believe that the past, present and future are all fixed, because, according to Einstein’s special theory of relativity, ‘now’ is totally subjective. This is the so-called ‘block universe’, which is a logical consequence of treating time as a spatial dimension, giving us space-time.

 

You can observe time as a dimension by looking at the night sky and seeing stars hundreds, if not thousands of years, in the past. This means that hypothetical observers in different parts of the Universe see a different ‘now’ and will observe events occurring in different sequences. This is a logical consequence of the finite speed of light. However, causally related events must happen in an objective sequence, irrespective of observers. This is Type B time, as defined by McTaggart. We are able to deduce causality of events that have happened in our past, which gives us theories of cosmology and evolution. This has to be compatible with Type A time, which is dependent on the fact that we all live in the present all of the time. 

 

Whether our present is different to someone else’s present (somewhere else in the Universe) just means that their Type A experience of time is different to ours, but Type B time occurs regardless of conscious observers.