Lesson of the Monty Hall Problem

On the television game show Let’s Make a Deal, Monty Hall, the show’s best known host, used to present contestants with the following situation: the contestant would be presented with three doors behind one of which was a big prize (say a brand new car). Behind the other two doors was a small prize (say $100 in cash). The contestant got to choose one of these doors. If the contestant chose the correct door (i.e., the one with the big prize), she could keep it.

But to make things interesting, after the contestant chose a door, Monty Hall would open one of the other two doors, making sure that it was one with a small prize ($100 in cash). Hall could always find such a door because only one had a big prize and the other two had small prizes—so there would always be such a door regardless of whether the contestant chose the correct door.

Now, after Hall opened a door with a small prize, he gave the contestant the following option: she could stay with her initial choice of doors or switch to the other unopened door. Once she decided whether to stay or switch, all doors would be opened and the contestant would only receive the big prize if it was behind the door she had finally settled on. So the Monty Hall problem is this: should she stay with her original door or switch?

There’s been a huge amount written on this problem and many smart people, when initially presented with it, think it doesn’t matter whether the contestant stays with her initial door or switches. Regardless of the contestant’s choice of doors, they reason, Hall knows what’s behind all three doors and can always open one with a small prize behind it. People seem psychologically programmed to think it doesn’t matter whether to stay or switch doors.

And yet, it turns out that switching rather than staying with one’s initial door is a much more effective strategy for winning the big prize. Indeed, by switching you increase your chances of winning the big prize from 1/3 to 2/3. Why is this?

Think of it this way: Initially you know nothing about which door has the big prize behind it. You’re probability of getting the big prize is therefore 1/3. Moreover, if your strategy is to stay with your initial choice, your probability doesn’t change—in that case, it doesn’t matter whether Hall subsequently opens one of the doors.

But now think of what happens when you switch. Initially you had a 1/3 probability of choosing the right door. If you did happen to choose the right door initially, now switching loses you the big prize. So you now have a probability of 1/3 of not getting the big prize. But that means your probability of getting the big prize, by switching, is now 2/3. It’s gone up!

In a sense, with the switching strategy, you hope that your initial choice is wrong so that by switching you get it right. But initially, your chance of getting it right is only 1/3 and getting it wrong is 2/3. By switching, your chance of success is the same as getting it initially wrong, so your probability of getting it right has gone up to 2/3. Hence, if you ever appear as a contestant on Let’s Make a Deal, choose the switching strategy.

Of course, it’s unlikely that you’ll ever appear on Let’s Make a Deal. So what’s the take-home lesson from the Monty Hall problem that you can apply to your life? It’s this: whenever there’s some event that’s important to you and you get new information about it, this new information can change the event’s probability or perhaps even prompt a course of action on your part that changes the event’s probability.

If it’s an event you want to avoid, then you’ll want to bring the probability down. If it’s an event you want to have happen (as in winning the big prize), then you’ll want to bring the probability up. In any case, you’ll want to do the probability calculation and see precisely how the probability has changed in light of the new information.

The Monty Hall problem is counterintuitive in that people’s first reaction tends to be that it doesn’t really matter whether to switch or stay with one’s initial choice. In these situations, it’s therefore not enough trust one’s intuitions. One needs, rather, to get out paper and pencil and do the calculation.

Even extensive training in statistics tends not to hone our intuitions in situations like the Monty Hall problem. Indeed, trained statisticians do no better than the rest of us until they apply their training and calculate the precise probabilities in question. The Monty Hall problem thus shows how our intuitions can mislead us and how mathematics can correct those intuitions.



Edward J. Barbeau, Mathematical Fallacies, Flaws and Flimflam (Washington, DC: The Mathematical Association of America, 2000), 86–90.

Amos Tversky and Daniel Kahneman, “Judgment under uncertainty: Heuristics and biases,” Science 185 (1974): 1124–1131.


Goedel’s Theorem for Dummies

When people refer to “Goedel’s Theorem” (singular, not plural), they mean the incompleteness theorem that he proved and published in 1931. Kurt Goedel, the Austrian mathematician, actually proved quite a few other theorems, including a completeness theorem for first-order logic. But the incompleteness theorem is the one for which he is most famous.

To get some sense of the impact of Goedel’s Theorem on the mathematical community, consider how Herman Weyl, perhaps the greatest mathematician of the first half of the twentieth century, reacted to it. According to Weyl, up until Goedel’s Theorem, mathematicians were optimistic that all questions in mathematics could be definitively answered either one way or another (see his contribution to Oldenbourg’s Handbook of Philosophy).

Thus, for any mathematical claim, they thought a proof of either it or its negation exists. Goedel’s Theorem showed that this was not the case. Take the famous Goldbach Conjecture, which states that any even number greater than 2 can be written as the sum of two prime numbers (prime numbers are divisible only by themselves and one). To date, mathematicians have been able to find such a pair of prime numbers for any even number greater than 2 that they’ve considered. But they have not been able to prove that this result holds for all even numbers.

Perhaps the Goldbach’s Conjecture is false. But to show that, we would have to produce an even number greater than two that is not the sum of prime numbers. Perhaps it is true. In that case we’ll never find an even number greater than two that is not the sum of two prime numbers, but to actually prove this we cannot simply run through all even numbers, showing that the result holds in each case—that would require checking an infinite number of cases, and proofs in mathematics, some of which are thousands of pages long, nonetheless always consist of a finite number of steps.

What, then, about Goldbach’s Conjecture—does it or its negation admit a strict proof? Prior to Goedel’s Theorem, mathematicians thought that it did. Afterward, they could no longer be sure. Afterward, it might be that Goldbach’s Conjecture or its negation could be proved. But it could also be that Goldbach’s Conjecture was true but not provable. Or it might be false, but its falsehood was not provable.

Goedel’s Theorem shows that mathematical truth and mathematical proof are not the same thing (if they were, Goedel’s Theorem would have been a completeness theorem demonstrating the identity of mathematical truth and proof). Truth is about the way things are; proof is about what we can know to be true. Goedel’s Theorem shows us that there are many claims in mathematics that are true but whose truth we cannot know, at least not by mathematical proof (perhaps God can simply tell us that the claim is true, in which case we might have good reason to believe it, but we still wouldn’t know that it is true in the conventional mathematical sense of having a proof).

In mathematics, we consider ourselves as knowing the axioms of a given mathematical theory—these are usually taken to be self-evident truths and they form a manageable set in the sense that we know whether any claim is or isn’t an axiom. Besides the axioms, we also consider ourselves as knowing the claims that follow via the rules of logic from those axioms. The axioms and their logical consequences are the claims that we can prove. Goedel’s Theorem states that there are always truths that are not knowable in this sense.

Now one might say, well, if there are such truths, why don’t we just add them to our axioms. If we keep doing this, maybe we can render all mathematical truths provable in that way. But this strategy doesn’t work because Goedel’s Theorem shows that every time we add a new axiom, there’s a new “Goedel Sentence” that is true but not provable from the new axiom system.

The key to proving Goedel’s Theorem is constructing what has come to be called a Goedel Sentence, which asserts (and the assertion is provably true!) that it is not provable within the axiom system under consideration. The Goedel Sentence is a self-referential statement, much like “This sentence is false.” By accurately asserting of itself that it is not provable, it ends up being true but not provable. This is a mindbender, but it works, and it shows that there will always be true mathematical statements that we cannot prove. This is the upshot of Goedel’s Theorem.


Americans at Work: The Best and Worst Jobs


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Americans at Work: The Best and Worst Jobs

Most Americans spend more time on the job than they do on anything else.


The average employee spends more than 2/3 of his or her day at work or on work-related activities. That’s more time than we spend sleeping or raising our children.

Americans work an average of nearly one month more per year now than in 1970.

In 1960, only 20 percent of mothers worked. Today, in 70 percent of American households all adults work.

America vs. the world:

  • Americans work 137 more hours per year than Japanese workers
  • 260 more hours per year than British workers
  • 499 more hours per year than French workers
  • Average productivity for American workers has increased 400% since 1950
  • In every country included except Canada and Japan (and the U.S., which averages 13 days/per year), workers get at least 20 paid vacation days. In France and Finland, they get 30 – an entire month off, paid, every year.

So it matters what you do… doesn’t it? Because Americans work so much….

Here are the 10 Best AND 10 Worst Jobs in America, 2013 (with median salaries)

10 Best Jobs

#1 Actuary:

Salary: $91,211
Projected Job Growth: 27%
Unemployment rate: 0
What’s good: Job security, challenging, evolving job
What’s bad: Some consider it a lonely and boring job

#2 Biomedical Engineer

Median Salary (as of 2010): $85,163
Projected Job Growth: 62%
Unemployment rate: 0.4%
What’s good: Healthcare a growing field
What’s bad: You will be a jack of all trades, master of none

#3 Software Engineer

Salary: $89,147
Projected Job Growth: 30 %
Unemployment rate: 2.2%
What’s good: Companies need software engineers
What’s bad: Pressure job

#4 Audiologist

Salary: $68,135
Projected Job Growth: 37%
Unemployment rate: .4%
What’s good: Hearing aid technology improving
What’s bad: Very stressful

#5 Financial Planner

Salary: $64,750
Projected Job Growth: 32%
Unemployment rate: 2.4%
What’s good: Retiring baby boomers will need assistance
What’s bad: Stressful, at the mercy of the ups and downs of the stock market

#6 Dental Hygienist

Salary: $68,250
Projected Job Growth: 38%
Unemployment rate: .9%
What’s good: Opportunities in the field are abundant
What’s bad: Requires competitive tasks

#7 Occupational Therapist

Salary: $72,320
Projected Job Growth: 33%
Unemployment rate: .5%
What’s good: Increasingly caters to the aging boomer population
What’s bad: They cannot fully help someone recover from an injury

#8 Optometrist

Salary: $94,990
Projected Job Growth: 33%
Unemployment rate: .3%
What’s good: Opportunities with aging boomer population
What’s bad: Repetitive job, no career progression

#9 Physical Therapist

Salary: $76,310
Projected Job Growth: 39%
Unemployment rate: 1.8%
What’s good: Opportunities abound with older generations
What’s bad: Graduate education needed, licenses need to be renewed every two years

#10 Computer Systems Analyst

Salary: $77,740
Projected Job Growth: 22%
Unemployment rate: 3.2%
What’s good: Network administration is a critical component of business practices
What’s bad: Sitting at a desk all day watching monitors…if that’s OK with you, this is not a bad thing.

10 Worst jobs in America

#1 Newspaper Reporter

Salary: $36,000
Projected Job Growth: -6% (minus 6 percent)
Unemployment rate: 7.7%
What’s good: Informing the public
What’s bad: high stress, low pay, jobs disappearing

#2 Lumberjack

Salary: $32,870
Projected Job Growth: 4%
Unemployment rate: high (no accurate figures available)
What’s good: The great outdoors
What’s bad: Danger of working with heavy machinery in remote locations

#3 Enlisted Military Personnel

Salary: $41,998 (eight years experience)
Projected Job Growth: Varies
Unemployment rate: 0
What’s good: Serving your country
What’s bad: High stress, dangerous situations

#4 Actor

Salary: $17.44 an hour
Projected Job Growth: 4%
Unemployment rate: 8.6%
What’s good: It’s a calling, and if you make it big….
What’s bad: Very competitive, high stress

#5 Oil Rig Worker

Salary: $37,640
Projected Job Growth: 8%
Unemployment rate: 3.5 %, if you include the boomtowns in North Dakota
What’s good: Growth area in natural gas
What’s bad: Isolated at sea, long hours

#6 Dairy Farmer

Salary: $60,750
Projected Job Growth: -8%
Unemployment rate: 15.9%
What’s good: Independence, producing a product people need
What’s bad: Smaller farmers being forced out of business

#7 Meter Reader

Salary: $36,400
Projected Job Growth: -10%
Unemployment rate: 9%
What’s good: Outdoors work
What’s bad: Declining profession due to remote reading

#8 Mail Carrier

Salary: $53,090
Projected Job Growth: -26%
Unemployment rate: about 14%
What’s good: While USPS jobs shrinking, UPS jobs still available
What’s bad: Job becoming obsolete

#9 Roofer

Salary: $34,220
Projected Job Growth: 18%
Unemployment rate: 14%
What’s good: Outdoors work appealing to some people
What’s bad: Long hours spent in heat or cold. Work is cyclical.

#10 Flight Attendant

Salary: $37,740
Projected Job Growth: 0
Unemployment rate: 4.5%
What’s good: See the world
What’s bad: High stress, shrinking job market

BONUS: 5 Best jobs for people who hate (working with other) people


Job prospects through 2018: 22 percent increase according to Bureau of Labor Statistics.

Average Salary for Accountants/Auditors: $59,430*

Computer Programmer

Good News: Job opportunities are expected to soar 22 percent through 2018.

Average Salary for Computer Programmers: $69,620*

Forensic Science Technician

Good News: expected to grow 20 percent through 2018.

Average Pay for Forensic Science Technicians: $23.97/hour*

Budget Analyst

Good News: 15 percent increase in jobs predicted through 2018

Average Salary for Budget Analysts: $65,320*


Good News: Employment expected to jump 21 percent through 2018.

Average Salary for Actuaries: $84,810*




A World Without the Post Office


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A World Without the Post Office

The U.S. Postal Service, one of the few government agencies explicitly authorized by the U.S.
Constitution, has seen better days. After a plan by the post office to end most Saturday mail delivery,
a proposed federal budget mandated six-day delivery, despite post office officials’ contention that the
independent federal agency needs to trim costs. With steadily declining revenue, the specter of a world
without the U.S. Postal Service becomes more plausible each year.

The Size and Reach of the Post Office:


Postal Service-managed retail offices


Vehicles, one of the largest civilian fleets in the world

1.3 billion

Miles driven each year by letter carriers and truck drivers


World’s mail volume handled by the Postal Service

8 million



Pieces of mail processed every day

$65 billion

2012 revenue

$1.8 billion

Salaries and benefits paid every two weeks

423 million

Annual visits to

5.7 million

Passport applications accepted every year


Tax dollars received for operations

152 million

Total delivery points

Blame It on Gmail? Mail Volume Falls

With easy, free access to email, and thus email marketing, the demand for mail has fallen over the past

2 in 3

Americans have access to email; that is expected to rise to 72% by 2017

More Than Mail: Our Love Affair With Stamps

The fare for sending a letter—the stamp—is a huge part of U.S. culture. We celebrate with holiday-
themed stamps; we raise money with charity stamps; and we collect old stamps, well, because they’re


Stamps sold slightly above the cost of a regular stamp, semi-postal stamps raise funds for causes
identified by Congress.

Commemorative Stamps

The post office has released dozens of commemorative stamps over the years, whether related to
music, entertainment or Americana. Here are the most collected commemorative stamps.


U.S. Postal Service



Rise of the New LinkedIn: By the Numbers

The New LinkedIn

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Rise of the New LinkedIn: By the Numbers

You used to only check LinkedIn for job listings every few months—so why are working professionals suddenly spending hours on it every week.

LinkedIn’s New Numbers

Worldwide Growth

2003: 500,000 users
2007: 13 million users
2011: 140 million users
2017: 500 million users*
10+ million active job listings
9+ million companies
100,000 articles published weekly

LinkedIn usage in the United States

138 million members
3 million jobs posted every month
20,000 companies use LinkedIn for recruiting

Who Logs On?

In 2016, 29% of online adults used LinkedIn

LinkedIn user characteristics (out of online adults worldwide)


50% of college grads
27% of those with some college
12% of high school grads or less


45% of those with household income of ≥$75,000
21% of those with income of ≤$30,000

Employment status

35% of employed people
17% of unemployed people

Deeper, More Frequent Engagement

In 2017, worldwide active users* are growing
January: 227 million
February: 240 million
March: 260 million
20% increase in content consumption

“I read interesting articles/videos on my LinkedIn feed”

2014: 16%
2016: 36%

The reason for this jump in activity? People don’t log in just to update their resumes anymore.

New Reasons for Using LinkedIn

Before: Job Hunt Only

In August 2017, in the U.S. there were 7.1 million unemployed people
4.4% unemployment rate
35% of job seekers use social networks to contact potential employers
24% use them to contact recruiters or headhunters
75% of people who recently changed jobs use LinkedIn while making their next career decision

Now: Networking and Personal Branding

79% of professionals say networking is valuable for career growth
61% of professionals say regular online interaction with networks can lead to job opportunities
35% say a casual conversation on LinkedIn Messaging has led to a new opportunity
50% of LinkedIn members have found a job through a mutual connection

Most average LinkedIn connections per user
United Arab Emirates: 211
Netherlands: 188
Singapore: 152
London: 307
Amsterdam: 288
San Francisco Bay Area: 241
Staffing and recruiting: 702
Venture capital and private equity: 423
Human resources: 380

Expert Opinions on the New LinkedIn

Manu Goswami, Business Development Associate, JB Fitzgerald Venture Capital
“When I joined LinkedIn, no one was posting anything related to their professional life in a way that truly utilized the beauty of storytelling. When people started doing that, LinkedIn became an outlet for many people. It’s become a community I can count on for everything.”

Robyn D. Stoller-Shulman, Senior Editor, 51Talk
“LinkedIn was always a phenomenal place for employed professionals to network. However, recent changes to the platform encourage ongoing engagement through status updates, video, immediate feedback, and viral views.”

Thomas Ma, Co-Founder, Sapphire Apps
“If you’re starting out, don’t try to write something for the sake of the vanity. When you’re shooting for numbers, you probably won’t reach a lot of people. Be yourself and share your perspective. That’s the best way to build an audience.”

LinkedIn is Changing Along with Its Users

Focus on Making Connections
New Features
Native video
Share in-person status updates, tutorials, and behind-the-scenes videos on the LinkedIn mobile app
Active status
See when connections are online and ready to chat
Start informal conversations with Facebook-style instant messaging
Video filters
Apply Snapchat-style filters to videos taken through the LinkedIn app at conferences and conventions

What to watch for in the future

Mentoring: Free service would match LinkedIn members with seasoned pros who give career advice
In 2017: Limited launch in San Francisco and Australia
Access to Top Thought Leaders

Jeff Weiner CEO, LinkedIn
Posts about: business, newest innovations from around the world
Followers: 6,633,664

Liz Ryan, Founder & CEO, Human Workplace
Posts about: company improvement, landing your dream job
Followers: 2,556,53

Mohamed El-Erian, Chief Economic Advisor, Allianz
Posts about: global economy, financial markets, monetary policy
Followers: 1,476,151

LinkedIn’s Effectiveness

According to a 2016 Jobvite report, 87% of recruiters use LinkedIn to evaluate candidates
Employees sourced through LinkedIn are 40% less likely to leave the company within the first 6 months

LinkedIn hiring rate*
August 2016: 1.22
August 2017: 1.31
Up 7.2%

With all of the growth and updates, it looks like the new LinkedIn is here to stay!

The New LinkedIn



The Amazing History of Information Storage: How Small Has Become Beautiful

People have been storing information since the stone ages, ever since they’ve been writing or putting art on tablets and walls. With the invention of paper and ink, the “density of information” increased significantly, packing a lot more information into a tighter space (such scrolls and eventually bound books, as we still use today).

The invention of printing didn’t substantially increase the density of information, though it greatly contributed to its dissemination by making information easier to copy. In the 20th century, the benchmark for a sizable chunk of information became the Encyclopedia Britannica.

The 2010 edition (the last print-edition that Encyclopedia Britannica will ever publish) consists of 32 volumes and weighs 129 pounds. (Source) At 50 million words or about 300 million characters, it requires roughly gigabyte to store the text electronically (leaving out images and diagrams).

In an age of thumb drives that weigh less than an ounce and that, these days, routinely hold 8 or 16 gigabytes, paper and ink doesn’t seem like a very efficient way to store information. But electronic storage wasn’t always so efficient.