statistics

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.

The Monty Hall gameshow Three Doors Problem. There is a car behind one door, and goats behind the other two. You pick a door. Monty Hall, the gameshow host, opens one of the other doors and it has a goat. Should you change your choice of door? Yes, you should. But why? – Monty Hall Problem Stock Illustration ID: 1881849649 by SATYA94.

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 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 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.

To see the other Super Facts click here

US Violent Crime Nearly Cut in Half Since 1990

Superfact 8 : US Violent Crime Nearly Cut in Half Since 1990

Despite all the news reports about rampant crime, the US violent crime rate has fallen to half of what it was in the early 1990’s.

I’ve known for a while that violent crime in the US has been going down substantially since 1990. However, quite often when I mention this people refuse to believe it. When I visit NextDoor, the hyperlocal social networking service for neighborhoods, I see people complaining about rising crime, and especially rising violent crime. The news media and the newspapers are filled with violent crime stories, murders, mass shootings, assaults, rape, and robberies. It seems to be getting worse and worse. We are living in scary times, aren’t we?

The Better Angels of our Nature

A couple of years ago I bought a book by Steven Pinker with the title The Better Angels of our Nature, why violence has declined, which contradicted the violence is getting worse narrative. In the book he claims that violence is trending down worldwide and that includes US violent crime.

In 1987 I received a Christmas present from the parents of a fellow student whom I had been tutoring. I was an exchange student to the United States from Sweden at the time. The Christmas present in question was the World Almanac of 1987. I loved it and ever since I’ve bought the World Almanac every year. If you open the pages for crime statistics in the United States in the World Almanac you see the same thing, violent crime in the US is declining.

The World Almanac and Book of Facts 2023.

Below is what the World Almanac 2023 says about the violent crime rate per 100,000 residents in the United States (page 114). The data sources were : Federal Bureau of Investigation, U.S. Department of Justice, Uniform Crime Reporting Program, Crime in the United States 2020. I also included property crimes in the last column to show that non-violent crimes have declined as well.

Year	All violent crime	Murder and nonnegligent manslaughter	Rape	Robbery	Aggravated assault	All property crime (non-violent)
1990	729.6	9.4	41.1	256.3	422.9	5,073.1
1995	684.5	8.2	37.1	220.9	418.3	4,590.5
2000	506.5	5.5	32.0	145.0	324.0	3,618.3
2005	469.0	5.6	31.8	140.8	290.8	3,431.5
2008	458.6	5.4	29.8	145.9	277.5	3,214.6
2011	387.1	4.7	27.0	113.9	241.5	2,905.4
2014	361.6	4.4	26.6	101.3	229.2	2,574.1
2016	386.6	5.4	30.0	102.9	248.3	2,451.6
2017	383.8	5.3	30.7	98.6	249.2	2,362.9
2018	370.4	5.0	31.0	86.1	248.2	2,209.8
2019	368.6	5.1	31.4	81.8	250.4	2,130.6
2020	387.8	6.5	27.6	73.9	279.7	1,958.2

The graph below is taken from this article from the Pew Research Center tells the same story. The graph. It shows that the US violent crime rate has nearly halved since the 1990’s. There is a slight uptick in crime for the years 2020 and 2021 but according to this PBS article the downward trend has continued in 2022 and 2023.

The four graphs show that both violent crime and property crime has declined since 1990. Click on the picture to see the Pew Research article it is taken from.

Since this is a surprising fact that some does not even want to acknowledge (in my experience) I consider this a super fact.

Do you feel it is hard to believe that violent crime in the US has been declining over the last few decades?

To see the other Super Facts click here

Poverty and child mortality has been sharply reduced worldwide

Superfact 7 : Poverty and child mortality has been sharply reduced worldwide.

Extreme poverty as well as child mortality has been sharply reduced the world over. The countries that are the worst-off today are still better off than the countries that were doing the best at the beginning of the 19^th century. Over the last 20 years extreme poverty and child mortality has continued to decline sharply.

In Hans Rosling’s book Factfulness, “Ten Reasons We’re are Wrong About the World – and Why Things Are Better Than You Think”, the author presented a quiz that he wanted the reader to take before reading the book. Below is the third question of the quiz. I should say that Hans Rosling posed this and other questions to thousands of people around the world.

In the last 20 years, the proportion of the World population living in extreme poverty has…

A. Almost doubled
B. Remained more or less the same
C. Almost halved

The correct answer is C. Almost halved. Around 7% of the quiz takers got the right answer. Around 5% in the United States got it right. That the proportion of the World population living in extreme poverty halved in 20 years is a true fact. This is also an important fact about the world. Despite that most people got it wrong. In fact, monkeys randomly picking answers would do better (33%). I did pretty well on this quiz, but since I was reading the book and the book’s title is “Ten Reasons We’re are Wrong About the World – and Why Things Are Better Than You Think”, I got some help just looking at the front cover. In any case, I think this fact qualifies as a super fact, as do the other fact below (There has been a steep decline in child mortality).

There has been a steep decline in extreme poverty

According to the World Bank and Our Data World in Data, an organization which makes data in databases easily accessible to public, extreme poverty went from almost 80% 200 years ago, 60% 100 years ago, about 45% 50 years ago, 34.3% 30 years ago, 23.6% 20 years ago, 8.4% in 2019, then it went up to 9% during the covid epidemic but it seems to be back down to a bit above 8% again. See the graph below as well as the two linked articles in this paragraph.

I should say that extreme poverty is (by the UN, World Bank, etc.) “a condition characterized by severe deprivation of basic human needs, including food, safe drinking water, sanitation facilities, health, shelter, education and information” and is currently defined to be below $2.15. In 2011 it was below $1.90.

This graph from Our World In Data shows a steep decline in extreme poverty over time. Click on the picture to visit the original article.

Other related statistics mentioned in Hans Rosling’s book is that the share of undernourished people went from 28% in 1970 to 11% in 2015. That is despite the earth’s growing population. Related to this is that cereal yield per hectare went from 1.4 ton in 1961 to 4 ton in 2014. This is due to modern agricultural techniques. The share of people with water from protected sources went from 58% in 1980 to 88% in 2015. This statistic was taken from WHO, UNICEF, the World Bank and FAO (the UN Food and Agriculture Organization).

There has been a steep decline in child mortality

According to Hans Rosling’s book the percent of children dying before their fifth birthday went from 44% in 1800 to 4% in 2016. To get a more complete picture see these articles from Our World In Data, article1, article2, as well as the graph below. For most of humanities existence virtually all societies suffered a child mortality rate between 40% and 60%, but that changed drastically over the last 100 years.