Superfactful – Page 2

Time Dilation Goes Both Ways

Super fact 38 : If two observers are moving compared to each other both will observe the other’s time as being slower. In other words, both observers will observe the other’s clocks as ticking slower. Time slowing down is referred to as Time Dilation. And this post is about how time dilation goes both ways.

A lot of people know that if someone moves very fast his clocks will run slower. That’s relativity. If someone speeds through space in a rocket ship, close to the speed of light his time will slow down. When one hour passes on earth only half an hour may pass in the rocket. What comes as a shock to many people is when they find out that the converse is also true. When one hour passes in the rocket only half an hour will pass on earth.

Clearly that looks like a contradiction, but there is an explanation. I consider this a super fact because it is so strange and almost impossible for people to believe, and yet it is true.

The image shows two clocks side by side. On the left is a wall clock and on the right a wristwatch | Time Dilation Goes Both Ways — The guy on earth says my clock (left) is ticking double as fast as the rocket man’s clock (right). The rocket man say’s my clock (right) is ticking double as fast as the clock on earth (left). Who is right? Surprisingly both of them.

Postulates of Special Relativity

The two postulates of special relativity are:

The laws of physics are the same in all inertial frames of reference. An inertial frame is a system that moves at a constant velocity.
The speed of light in a vacuum is constant for all observers, regardless of the motion of the light source.

The first postulate is called the principle of relativity and goes all the way back to Galileo Galilei. It means that no experiment can determine whether you are at rest or moving at a constant velocity. The reciprocity of time dilation follows from this postulate. If the time for the rocket man in the example above was ticking at half the speed compared to the time for the guy on earth and they both agreed, then you could tell who was standing still and who was moving from that fact.

The first postulate demands that they disagree. The guy on earth thinks the rocket man’s clock is ticking at half the speed of his own clock, whilst the rocket man think it is earth man’s clock that is going slow. Therefore, you can’t tell who is standing still, which is what the first postulate requires.

The second postulate is the more shocking one and is special to relativity. It was discovered experimentally at the end of the 19^th century but was too difficult for scientists to accept at first so various ad hoc explanations were put forth to explain it away, until the theories of relativity were created. I designated this postulate as my super fact #4 and you can read about it here.

The picture shows two people Alan and Amy. Alan is on the ground. Amy is flying by Alan in a rocket speeding left. Both Alan and Amy are pointing lasers to the left | Time Dilation Goes Both Ways — In this picture Amy is traveling past Alan in a rocket. Both have a laser. Both measure the speed of both laser beams to be c = 299,792,458 meters per second. The speed of light is a universal constant.

Time Dilation

In the pictures below I am showing two rocket systems in space, Amy’s rocket and Alan’s rocket. They are travelling at a high speed compared to each other. Each rocket has a light clock that consists of a light beam bouncing up and down between a mirror in the ceiling and a mirror on the floor. The two light clocks are identical, and each bounce corresponds to a microsecond.

Amy is passing Alan at a high speed, and therefore Alan will see Amy’s light clock running slower than his because Amy’s light beam must travel further. Remember, the speed of light is identical for both light clocks (light speed is a universal constant). For those interested I am also deriving the formula for time dilation.

The picture shows two systems, each with a clock consisting of light beams bouncing between mirrors. In this set up Alan is stationary compared to us and therefore his light beam only moves vertically. — Alan and Amy have identical light clocks. We call the time it takes for the light beam to go from the floor to the ceiling (one clock tick) Dt in Amy’s case and Dt’ (reference frame) for Alan. Amy is speeding past Alan towards the left. From Alan’s perspective Amy’s clock is running slower. Using Pythagoras theorem, it is possible to derive the formula for time dilation shown in the lower left corner.

Since Amy moving left is the same as Amy standing still and Alan moving right you can say that Alan is the one moving fast. In this case it is Alan’s light clock that is ticking slower because from this viewpoint it is his light beam that has to travel further. From Amy’s perspective it is Alan’s clock that is going slower.

The picture shows two systems, each with a clock consisting of light beams bouncing between mirrors. In this set up Amy is stationary compared to us and therefore her light beam only moves vertically | Time Dilation Goes Both Ways — It is equally correct to say that Amy is standing still and that it is Alan that is moving fast to the right. This time (pun not intended) it is Alan’s clock that is ticking slower. Dt corresponds to Alan’s clock ticks and Amy’s clock ticks are Dt’.

This seemingly contradictory situation is resolved by the fact that Amy’s and Alan’s perspectives will drift apart as they continue their journey. They will increasingly disagree on whether events are simultaneous or not, and they will disagree in which order events occur. This is another shocking fact, or as I refer to it, super fact. It is strange but it resolves the apparent contradiction of reciprocal time dilation. I am explaining this in greater detail in this post.

The Twin Paradox

But what happens if one of Amy or Alan decides to turn around so that they meet up again. If Amy’s clock runs slower from Alan’s perspective and Alan’s clock runs slower from Amy’s perspective, how can you reconcile that when they meet up again? It turns out that whoever is turning around or accelerating or decelerating to turn back is the one who will have the least time pass. If Amy is the one turning back, then she will age less than Alan. During her acceleration she will see Alan’s clock starting to run faster and faster until he is older her.

Let say Alan’s clock is running half the speed of Amy’s clock from Amy’s perspective and Amy’s clock is running half the speed of Alan’s clock from Alan’s perspective. Let’s also say that Amy traveled to the left for 10 years before turning around.

From Alan’s perspective she would have traveled 20 years before turning around. However, from Amy’s perspective 5 years would have passed on Alan’s clock. As she turns around Alan’s clock will run faster and catch up so that when they meet up again Amy will be aged 20 years, while Alan will be aged 40 years. That is 35 years of catching up for Alan’s clock from Amy’s perspective. Alan’s clock advanced 35 years from Amy’s perspective after Amy turned around. In the end Amy will be the younger one.

The picture shows Amy on the left turning around and Alan on the right. Text explains what happens | Time Dilation Goes Both Ways — Observe that the fast-forward advancement of Alan’s clock from Amy’s perspective happens only while Amy is in the process of turning around (accelerating / decelerating). Further, how fast the fast forward happens depends on the distance as well. Once Amy is traveling at a constant speed again (inertial frame) Alan’s clock will run slower again from Amy’s perspective.

A somewhat halting but OK analogy for the 35 years of catching up that happens on Alan’s clock from Amy’s perspective is when you turn a boat around on a wavy sea. As you are moving in the direction of the waves the waves will hit you much less often (if at all) but after you turn around and move against them the waves will hit your boat very frequently. Alan’s clock will run faster for Amy whilst she is turning around.

Book Recommendations on Relativity

Relativity Visualized by Lewis Carrol Epstein. This book does not require much education in physics and very little math, and yet it explains relativity very well.
Relativity, The Special and the General Theory, by Albert Einstein. In my opinion, not the best overall, but it is easy reading for the public and the chapter on the relativity of simultaneity is pretty good.
Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles by Robert Eisberg and Robert Resnick. An advanced but excellent explanation of special relativity in the appendix.
The Special Theory of Relativity by David Bohm. It is written for people with some physics education. However, it is one of the best books on Special Relativity that I’ve read.
Einstein’s Miraculous Year. This book features the translations of the five famous papers that Albert Einstein wrote in 1905 including “On the Electrodynamics of Moving Bodies”, which was his paper on Special Relativity. Not for the faint of heart but very interesting.

To see the other Super Facts click here

There is strong evidence for the Big Bang

Super fact 37: There is strong evidence for the Big Bang, and we know a lot about how the Universe evolved through time since the Big Bang.

Considering the evidence that has accumulated throughout the years for the Big Bang it is hard to deny it happened. In my experience very few people are aware of this evidence, and they are surprised to find out how much evidence there is and how many details we know about the evolution of the universe. It is easy to believe that scientists are guessing when you don’t know much about the evidence yourself. But they are not guessing. That is why I call this a super fact.

In addition, there’s a lot of misconceptions around the Big Bang as well surprising facts. A few additional things that might surprise people are that the Big Bang was not like an explosion, the Universe did not expand into something. In addition, there might be multiverses and multiple Big Bangs, and there are cyclic models, and so-called eternal inflation. There are things we know and things we don’t know.

The pictures show an expanding Universe starting with quantum fluctuations followed by inflation, then an afterglow light pattern 375,000 after the Big Bang and then the so-called dark ages, the creation of stars and galaxies | There is strong evidence for the Big Bang — This file is in the public domain in the United States because it was solely created by NASA. (from Wikimedia commons

The Expansion of the Universe

It used to be believed that the Universe was static. In 1929 the astronomer Edwin Hubble discovered that the universe was expanding. He made this observation by analyzing the light from distant galaxies and noticing that their light was redshifted. I am going to explain what that means next.

Light emitted from elements, atoms and molecules have light absorption patterns that are unique to the atom/element in question. This is called a light spectrum. This makes it possible to identify the elements in a star and their proportions. Red shifted means that the absorption lines have moved towards red because the frequency of the light has been shifted due to the motion. This is called the doppler effect.

You can notice this phenomenon for the case of sound when an ambulance is coming towards you and then speeding by you. The sound changes. Hubble was using the redshift to the determine that further away the galaxy was the faster it was moving away from us.

The top shows a colorful spectrum from blue to red with absorption lines in black. The bottom portion of the picture shows the same thing expect the black absorption lines have moved a bit to the right. — Visualization of redshifted absorption lines are redshifted due to velocity away from observer. Top lines are for an object at rest and in the bottom picture the object is moving away. Maxmath12, CC0, via Wikimedia Commons. This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.

A natural explanation for this is that the universe is expanding, and that it once must have been much more compressed, but it is not the only explanation. However, there is more evidence.

The Cosmic Microwave Background Radiation

If you assume that the universe once was much more compact and much hotter than today, particles such as protons, neutrons and electrons would have been free and close together preventing light from freely moving around. However, as the universe kept expanding and cooling these particles eventually should have been able to form atoms allowing light or electromagnetic radiation to freely move around.

Some physicists, Alpher, Herman and Gamow predicted around 1950 that this should have left behind a detectable microwave background radiation. This radiation was detected by chance in 1964 by two physicists, Penzias and Wilson. This radiation had the expected properties and careful study of this Cosmic Microwave Background Radiation (CMBR or CMB) has given us a lot of information about our universe and may give us information about other universes (multiverses).

Its existence is strong evidence that the universe once was very compressed and much hotter, i.e., the Big Bang. An interesting fact is that in old TVs, between the channels (old people will remember this), you had this fuzz, or war of the ants as some people called it, and part of that TV fuzz is the CMBR.

A big sky map with varying colors, yellow, red, green and blue | There is strong evidence for the Big Bang — This is a sky map of the cosmic background radiation from when the universe was around 380,000 years old. It was created with the help of satellite (NASA) measurements. The colors are artificial and show tiny temperature variations. NASA / WMAP Science Team, Public domain, via Wikimedia Commons.

You can watch a 4-minute video about the discovery of the CMBR narrated by Neil DeGrass Tyson on this PBS web page by clicking here.

Abundances of Light Elements

Yet another piece of evidence is the relative abundance of hydrogen and helium compared to heavier elements. The physics at the beginning of the Universe under the Big Bang tells us that initially regular matter should have consisted of 75% hydrogen and 25% helium and hardly anything else, and that is composition the oldest stars had when they were new. In addition, the oldest stars we’ve found appear to have an age just under the 13.8 billion years that we get for our Universe assuming the Big Bang theory (that’s another piece of evidence).

The picture consists of two pie chart graphs representing stars. The left one is a first-generation star with one pie for the 75% hydrogen and one pie for the 25% helium. — The first-generation stars consisted of 75% hydrogen and 25% helium and trace amounts of Lithium. A second or third generation star like our sun is still mostly hydrogen and helium but also many other elements. The rocky planets circling the sun are mainly elements heavier than hydrogen and helium. Image credit NASA, ESA, CSA, STScI

A Very Brief History of Time

So, it all started with a quantum fluctuation. The first 0.000000000000000000000000000000001 seconds is called the inflationary period characterized by super-fast expansion, much faster than the speed of light, as we know it today. At this time the strong nuclear force becomes distinct from the weak nuclear force.

I should point out that during the first 0.0000000000001 seconds the physics laws may not have applied in a normal sense. I should also point out that this was not an explosion. An explosion explodes into something but there was nothing else outside of the universe, so this is more like superfast growth.

At a fraction of a second protons and neutrons form from quarks and after one second neutrinos came into existence and if primordial black holes exist, they were formed at this time too. After two minutes nucleus consisting of neutrons and protons are formed, and the first elements hydrogen and helium formed. After 20 minutes an opaque hot plasma forms, after 100,000 years neutral helium atoms form, and after 375,000 years CMBR is created, etc.

This is just a small sample of everything that we know happened after the Big Bang, based on the known laws of physics. You can read about all the details in books like The First Three Minutes by Steven Weinberg or A Brief History of Time by Stephen Hawking.

vector illustration of up and down quarks in proton and neutron on white background. The proton (left) is a red and blue up quark and a green down quark. The neutron is a red and green down quark and a blue up quark | There is strong evidence for the Big Bang — The proton and neutron each consist of three quarks. They are formed at a fraction of a second after the Big Bang. Asset id: 2333679305 by KRPD.

Pre–Big Bang Cosmology

So, what was there before the Big Bang, if anything? Well, that part is speculation, but there are many good ideas. A popular hypothesis speaks of quantum fluctuations setting off the Big Bang. There are models in which the whole of spacetime is finite, including the Hartle–Hawking no-boundary condition. This means that time itself came into existence with the Big Bang and therefore nothing could have preceded it. This means that the Big Bang could not have been caused or created by anything else, just like a universe that has existed eternally could not have been caused or created by anything else.

In a sense, despite being 13.8 billion years old, the universe in this scenario would have always existed. Because the concept of “always” stops at 13.8 billion years ago. Stephen Hawking used the analogy of the north pole. You can’t go further north from the north pole.

There are other models that include multiverses, for example, eternal inflation, in which universal inflation ends locally here and there in a random fashion, each endpoint leading to a bubble universe, expanding from its own big bang. You can view this model as inflation being the river of time with the various universes popping up like bubbles in the stream.

In another model inflation is due to the movement of branes in string theory and Big Bangs are the result of colliding branes. There are cyclical models, such as Nobel Prize winner Roger Penrose’s Conformal cyclic cosmology in which one universe gives rise to another universe as it dies.

Notice that Pre-Big Bang Cosmology is speculative, but the reality of the Big Bang is backed by strong evidence.

Thousands of universes represented as colorful balls | There is strong evidence for the Big Bang — The surface of a multiverse with a lot of universes 3d rendering Asset id: 2256998119 by Dr. Norbert Lange.

To see the other Super Facts click here

Scientists Agree that Global Warming is happening and that we are the Cause

Super fact 34: Climate Scientists agree that Global Warming or if you call it Climate Change is happening, and that it is caused by us primarily because of our burning of fossil fuels. There is a long-standing scientific consensus on these two facts because the evidence is conclusive. Typically, studies show an agreement of at least 97% or 98% among climate scientists.

This is a super fact because surveys show that this is not what the public believes and yet it is true. The public incorrectly believes that there is a large disagreement among scientists on this topic. A note, to understand why the evidence is conclusive as to why global warming is happening and is caused by us click here.

Note : I will use the term “global warming” in this review. Whether you call the phenomenon climate change, climate disruption, or global heating, is not important.

The Scientific Consensus

This extensive survey from 2013 of 12,000 climate papers (papers published over two decades) by Dana Nuccitelli and Cook, etc., concluded that 97.1% of climate scientists endorsed the consensus position that humans are causing global warming.

They also did a science author self-rating which concluded that 97.2% of climate scientists endorsed the consensus position that humans are causing global warming. Another conclusion from the survey was that the consensus had increased from around 90%, perhaps less, in the early 1990’s.

A later review of six independent, peer-reviewed studies examining the scientific consensus about global warming have concluded that between 90% and 100% of climate scientists are convinced human-caused global warming is happening. A more recent study (2021) found that as many as 98% of climate scientists are convinced global warming is happening and is human-caused. Numerous other surveys have concluded the same thing.

People’s Beliefs About Global Warming

This 2024 survey from Yale University show that most Americans (61%) understand that global warming is mostly human caused. By contrast, 28% think it is caused mostly by natural changes in the environment. Most Americans (58%) <<Link-6>> understand that most scientists think global warming is happening. This percentage has trended generally upward since this survey began in 2008. By contrast, about one in five (22%) think there is a lot of disagreement among scientists about whether global warming is happening.

The green graph is going up slightly starting from 46% in 2009 and ending in 58% in 2023. The black graph starts at 33% in 2009 and ends in 22% in 2023. The yellow graph starts at 2% in 2009 and ends in 2% in 2023 | Scientists Agree that Global Warming is happening and that we are the Cause — The green graph corresponds to “most scientists think global warming is happening (%).” The black graph corresponds to “there is a lot of disagreement among scientists (%)”. The yellow graph corresponds to “Most scientists think global warming is NOT happening (%)”. Graph taken from the Yale Program on Climate Change Communication.

However, only one in five Americans (20%) understand that nearly all climate scientists (more than 90%) think that human-caused global warming is happening. The aforementioned Dana Nuccitelli refers to this in his book Climatology versus Pseudoscience as the consensus gap. Again, this large discrepancy between public perception and reality makes the consensus gap a super fact. Research has shown that this discrepancy has a large impact on people’s other beliefs regarding global warming.

This is bar graph. It shows that 2% believe the answer is 0-10%, 2% believe the answer is 11-20%, 3% believe the answer is 21-30%, 3% believe the answer is 31-40%, 8% believe the answer is 41-50%, 7% believe the answer is 51-60%, 7% believe the answer is 61-70%, 13% believe the answer is 71-80%, 13% believe the answer is 81-90%, 20% believe the answer is 91-100%, 22% don’t know | Scientists Agree that Global Warming is happening and that we are the Cause — The question was, To the best of your knowledge what percentage of climate scientists think that human-caused global warming is happening? Graph taken from the Yale Program on Climate Change Communication.

Why is there a Consensus Gap?

In his book Climatology versus Pseudoscience Dana Nuccitelli explains that a relatively small group of so-called climate skeptics, or more accurately called climate contrarians have received a lot of attention from media. Even though their science is bad, and they’ve published their error ridden papers in obscure or discredited journals, and the fact that their predictions have failed repeatedly many times over, they have had an enormous influence on public discourse. Conservative politicians, and many talk show hosts are blindly devoted to their falsehoods, whilst real scientists are being attacked.

It is not just rightwing media who are using them for their purposes, but mainstream media are giving the contrarians undue attention as well. Sensationalism is one issue. A science contrarian claiming that all the climate scientists are wrong, and that he is the only one who finally got it right is a lot more interesting of a story than a repeat of the consensus. Another issue is false balance. Journalist should not feel that they must give equal time to evidence-based science and nonsense, but that is often the case. To read my review of this book click here.

The Oregon Petition

I am mentioning the Oregon petition because I fell for it myself. The Oregon petition was an official looking petition circulated by climate contrarians, claiming that there is no evidence that human-caused global warming will cause catastrophic heating of earth’s atmosphere and disruption of earth’s climate, and that adding more carbon dioxide to the atmosphere would even be beneficial for plants and animals. It got an impressive number of signatures, 32,000 after some years.

However, it turned out that the signatories rarely had climate expertise, and were not scientists, and the survey listed many falsified names such as the names of the Spice Girls and several fictional characters. Less than 200 of the signatories were climate researchers.

It was a con, but it was touted in a lot of media as the truth. I saw it over and over and I believed it. I was later surprised to learn that it was a con and that a scientific consensus existed on global warming / climate change. Learning that I had been bamboozled on this matter was one of the red flags that prompted me to start doing some fact checking on the issue global warming.

To see the other Super Facts click here

Smallpox Killed 300 million People in the Last Century Before Eradication

Superfact 24: Smallpox killed 300 million people in the 20^th century. However, there have been no naturally occurring cases of smallpox since 1977, and the world was declared free of smallpox on May 8, 1980, by the 33rd World Health Assembly.

Vial with smallpox vaccine and syringe against blurred doctor's face. 3D rendering | Smallpox Killed 300 million People in the Last Century Before Eradication — Smallpox vaccine Stock Illustration ID: 1782022109 by Novikov Aleksey

300 million people is an astonishing number. It is six times the 50 million people who died from the Spanish flu. It is about four times as many people as the 70 to 85 million people who died in World War II. It is close to the entire current population of the United States. That’s how many people died from this very dangerous disease. It was eradicated by a vaccination campaign.

I think this fact qualify as a super-fact, first of all because of the astonishingly huge number of deaths but also for the fact that it is gone. It is hard to believe that this happened. It is hard to believe that the world has changed so drastically for the better. It is a shocking but true fact. Thanks to the vaccination campaign we are living in a much better world.

The picture is a world map showing countries in different colors. The colors indicated when smallpox was eradicated in that country. Dark blue indicates that it was before 1900 and that is Sweden and Norway. Light beige indicates it was eradicated in the 1940’s, which is true for the United States. Dark brown indicates that it was eradicated in the 1970’s and that represents, for example, India, Brazil and many African countries | Smallpox Killed 300 million People in the Last Century Before Eradication — This world map shows when smallpox was eradicated from different countries. The source is Our World in Data, originally Fenner et al. at CDC.

What is Smallpox?

Smallpox is an infectious disease caused by the variola virus<<Link-1>>. The last naturally occurring case was diagnosed in October 1977, and the World Health Organization (WHO) certified the global eradication of the disease in 1980. The disease begins with fever and vomiting followed by the formation of ulcers in the mouth and a skin rash that later turns into fluid filled blisters with a dent in the middle. These blisters get scabbed and leave scars. The death rate was about 30%.

A child with covered by severe blisters. — Child with smallpox in Bangladesh 1975. Wikimedia commons photo by CDC/James Hicks.

The Eradication of Smallpox and Vaccines

The smallpox vaccine has a long history that begin in China where smallpox inoculation had existed long before it did in Europe. In 1796 the English physician Edward Jenner demonstrated the effectiveness of cowpox to protect humans from smallpox. Soon after several countries enacted mandatory vaccinations.

In 1807, Bavaria became the first country in the world to introduce compulsory vaccinations. In 1958 the World Health Assembly was called upon to eradicate smallpox. At this point 2 million people still died from smallpox every year. In 1967 the World Health Organization intensified the global smallpox eradication. As mentioned, smallpox was eradicated at the end of the 1970’s.

In 1998 & 2002 vaccination was dealt a blow by the Wakefield studies claiming that the MMR vaccine caused autism. Even though the studies were debunked, and several later studies showed no link between the MMR vaccines and autism, the fear of vaccines began to spread.

For example, in 2024 the American Veterinary Medical Association reported 37% of the dog owners surveyed believe canine vaccination could cause autism in their dogs. Not only is there no link between vaccines and autism, but technically speaking, dogs cannot be autistic as the condition is unique to humans. Unfortunately, the unnecessary fear of vaccines causing autism seems to only be getting worse.

To see the other Super Facts click here

Are you vaccinated against smallpox?

The Surprising Monty Hall Problem

Superfact 22: Suppose you’re on a game show, and you’re given the choice between three doors: Behind one door is a car; behind the other two doors there are goats. You want to pick the car. You pick a door, and the host, who knows what’s behind the three doors, opens another door revealing a goat. Now the question is, is it to your advantage to switch door choice? The answer is yes. And that is the surprising Monty Hall Problem.

It is quite common to argue that it does not matter. You don’t know what is behind the two remaining doors so it should be 50/50 right? In a test involving 228 people only 13% chose to switch. However, you should switch.

Monty Hall, the gameshow host of the Let’s Make a Deal television game show, knows where the car is, so he never chooses the door with the car. And by curating the remaining two doors for you, he raises the odds that switching is always a good bet. By switching your choice, you have a 2/3 chance of winning the car but if you stay with your original choice, you only have a 1/3 chance of winning the car.

So why is this a super-fact? First, we know it is true. It is mathematically proven and experimentally verified that switching door is the best choice. Secondly, this was widely contested and is still surprising to people. Finally, probabilistic thinking is the key to being rational and making good decisions. This fact is true, important and disputed and thus a super fact.

One way of viewing the situation is by noting that there is a 1/3 chance that the car is behind any door that the contestant picks and a 2/3 chance that the car is behind one of the other two doors.

The picture shows three doors, one marked 1/3 and two more grouped together under 2/3 | The Surprising Monty Hall Problem — The car has a 1/3 chance of being behind the contestant’s pick and a 2/3 chance of being behind the other two doors. Picture from Wikimedia commons public domain.

If Monty opens one of the two doors that the contestant did not pick there is still a 1/3 probability that the car is behind the door the contestant picked and a 2/3 chance that the car is behind one of the other two doors. However, one of the doors that the contestant did not pick is now known to feature a goat. Therefore, the probability that the car is behind the other door is 2/3.

The picture shows three doors, one marked 1/3 and two more grouped together under 2/3. The last door has a goat, and it is marked by 0. The door in the middle is marked by 2/3. — The host opens a door. The odds for the two sets don’t change but the odds become 0 for the open door and 2/3 for the closed door. Picture from Wikimedia commons public domain.

The table below is probably (no pun intended) a better way of illustrating the situation. In the table door 1 is the door designated to be the contestant’s first choice. Monty opens one of the remaining doors that has a goat behind it.

Behind door 1	Behind door 2	Behind door 3	Result if staying at door 1	Result if switching to door offered.
Goat	Goat	Car	Wins goat	Wins Car
Goat	Car	Goat	Wins goat	Wins Car
Car	Goat	Goat	Wins Car	Wins goat

There are various other ways of explaining the situation including Steven Pinker’s approach. It is easy to test this is real life and repeated experiments and simulations shown that if you switch you have a 2/3 chance of winning.

As an example of the controversy this probability puzzle caused was Marily Savant’s column in Parade Magazine. As a side note, Marilyn Vos Savant is the person who has the highest recorded intelligence quotient (IQ) as stated in the Guinness Book of Records. In response to a question regarding the Monty Hall game show problem she wrote that you should switch. She received letters from 10,000 readers disputing this, including 1,000 with PhDs. In the long run she prevailed.