The probability puzzle that makes your head melt



A reference in a recent Magazine article to the Monty Hall problem - where a contestant has to pick one of three boxes - left readers scratching their heads. Why does this probability scenario hurt everyone's brain so much, asks maths lecturer Dr John Moriarty.

Imagine Deal or No Deal with only three sealed red boxes.

The three cash prizes, one randomly inserted into each box, are 50p, £1 and £10,000. You pick a box, let's say box two, and the dreaded telephone rings.

The Banker tempts you with an offer but this one is unusual. Box three is opened in front of you revealing the £1 prize, and he offers you the chance to change your mind and choose box one. Does switching improve your chances of winning the £10,000?

Marcus du Sautoy explains probability and the Monty Hall problem to Alan Davies on Horizon

Each year at my university we hold open days for hordes of keen A-level students. We want to sell them a place on our mathematics degree, and I unashamedly have an ulterior motive - to excite the best students about probability using this problem, usually referred to as the Monty Hall Problem.

This mind-melter was alluded to in an AL Kennedy piece on change this week and dates back to Steve Selvin in 1975 when it was published in the academic journal American Statistician.

It imagines a TV game show not unlike Deal or No Deal in which you choose one of three closed doors and win whatever is behind it.

One door conceals a Cadillac - behind the other two doors are goats. The game show host, Monty Hall (of Let's Make a Deal fame), knows where the Cadillac is and opens one of the doors that you did not choose. You are duly greeted by a goat, and then offered the chance to switch your choice to the other remaining door.

Most people will think that with two choices remaining and one Cadillac, the chances are 50-50.

The most eloquent reasoning I could find is from Emerson Kamarose of San Jose, California (from the Chicago Reader's Straight Dope column in 1991): "As any fool can plainly see, when the game-show host opens a door you did not pick and then gives you a chance to change your pick, he is starting a new game. It makes no difference whether you stay or switch, the odds are 50-50."

But the inconvenient truth here is that it's not 50-50 - in fact, switching doubles your chances of winning. Why?

Let's not get confused by the assumptions. To be clear, Monty Hall knows the location of the prize, he always opens a different door from the one you chose, and he will only open a door that does not conceal the prize.

For the purists, we also assume that you prefer Cadillacs to goats. There is a beautiful logical point here and, as the peddler of probability, I really don't want you to miss it.

In the game you will either stick or switch. If you stick with your first choice, you will end up with the Caddy if and only if you initially picked the door concealing the car. If you switch, you will win that beautiful automobile if and only if you initially picked one of the two doors with goats behind them.

If you can accept this logic then you're home and dry, because working out the odds is now as easy as pie - sticking succeeds 1/3 of the time, while switching works 2/3 of the time.

Kamarose was wrong because he fell for the deception - after opening the door, the host is not starting a new 50-50 game. The actions of the host have already stacked the odds in favor of switching.

The mistake is to think that two choices always means a 50-50 chance. If Manchester United play Accrington Stanley in the Cup then, with the greatest respect to proud Stanley, it's more likely that United will progress to the next round.

Still not convinced? You are in good company. The paradox of the Monty Hall Problem has been incredibly powerful, busting the brains of scientists since 1975.

In 1990 the problem and a solution were published in Parade magazine in the US, generating thousands of furious responses from readers, many with distinguished scientific credentials.

Part of the difficulty was that, as usual, there was fault on both sides as the published solution was arguably unclear in stating its assumptions. Subtly changing the assumptions can change the conclusion, and as a result this topic has attracted sustained interest from mathematicians and riddlers alike.

Even Paul Erdos, an eccentric and brilliant Hungarian mathematician and one-time guest lecturer at Manchester, was taken in.

So what happens on our university's open days? We do a Monty Hall flash mob. The students split into hosts and contestants and pair up. While the hosts set up the game, half the contestants are asked to stick and the other half to switch.

The switchers are normally roughly twice as successful. Last time we had 60 pairs in 30 of which the contestants were always stickers and in the other 30 pairs always switchers:

Among the 30 switcher contestants, the Cadillac was won 18 times out of 30 - a strike rate of 60%
Among the 30 sticker contestants, there were 11 successes out of 30, a strike rate of about 36%
So switching proved to be nearly twice as successful in our rough and ready experiment and I breathed a sigh of relief.

I had calculated beforehand the chances of ending up with egg on my face and the team of 30 stickers beating the 30 switchers. It was a risk worth taking, but one shouldn't play Russian Roulette too often.

Next year we will need something different, perhaps Simpson's Paradox. Imagine that 1% of people have a certain disease.

A diagnostic test has been developed which performs as follows - if you have the disease, the test has a 99% chance of giving the result "positive", while if you do not have the disease, the test has 2% chance of (falsely) giving the result "positive".

A randomly chosen person takes the test. If they get the result "positive", what is the probability that they actually have the disease? The answer, 1/3, is perhaps surprisingly low.


Click here for the article on the web.



<< Click here to see more recent news articles >>

 

 

Harnham blog & news

With over 10 years experience working solely in the Data & Analytics sector our consultants are able to offer detailed insights into the industry.

Visit our News & Blogs portal or check out our recent posts below.

Black History Month: Ethical AI and the Bias Within

According to Brigette Hyacinth’s 2017 book entitled, The Future of Leadership, the author suggests this when considering the ramifications of AI. “Using AI to improve efficiency is one thing, using it to judge people isn’t something I would support. It violates the intention on the applications of AI. This seems to be social prejudice masquerading as science…” How often have big tech companies backtracked their facial recognition software? What are the ethical implications of moving forward and leaving AI unchecked and unregulated? 2020 was in no way a traditional year amassing change on our daily lives at near lightspeed, or so it seemed. But what was brought to bear were unrest and tensions boiled to the breaking point. And when you look at it from the perspective of AI in our daily lives. What might the world look like in another year? When Social Sciences and Humanities Meets AI “To err is human, to forgive, divine.” Humans make mistakes. Biases are unmasked with and without intent. But, when it comes to AI, those unintentional biases can have devastating consequences. From 2015 to 2019, use of AI grew by over 250 percent and is projected to boast a revenue of over $100 billion by 2025. As major businesses such as Amazon and IBM cancel and suspend their facial recognition programs amidst protests against racial inequality, some realize more than regulatory change is needed. Since 2014, algorithms have shown biases against people of color and between genders. In a recent article from Time.com, a researcher showed the inaccuracies of prediction for women of color, in particular. Oprah Winfrey, Michelle Obama, and Serena Williams skewed as male. Three of the most recognizable faces in the world and AI algorithms missed the mark. These are the same algorithm and machine learning principles used to challenge humans at strategy games such as Chess and Go. Where’s the disconnect? According to one author, it may be time to create a new field of study specific to AI. Though created in Computer Science and Computer Engineering labs, the complexities of human are more often discussed in the field of humanities. To expand further as well into business schools, race and gender studies, and political science departments. How Did We Get Here? At first blush, it may not seem comparable to consider human history with the rise of artificial intelligence and its applications. Yet it’s human history and its social construct which explains the racial and gender biases when it comes to ethics in AI. How deep seated are such biases? What drives the inequalities when AI-enabled algorithms pass over people of color and women in job searches, credit scores, or assume status quo in incarceration statistics? Disparities between rational and relational are the cornerstone from which to begin. Once again, in Hyacinth’s book, The Future of Leadership, the author tells a story of her mother explaining the community around the simple task of washing clothes. Though washing machines now exist and do allow people to do other things while the clothes are washed, there is a key element recounted by her mother washing machines lack. The benefit of community. When her mother washed clothes, it was her and her surrounding community. They gathered to wash, to visit, and connect. A job was completed, but the experience lingered on. And in the invention of a single machine, that particular bit of community was lost. But it’s community and collaboration which remind humans of their humanity. And it’s from these psychological and sociological roles, artificial intelligence should learn. Create connections between those build the systems and those who will use them.  BUILDING AI FORWARD Voices once shuttered and subjugated have opened doors to move artificial intelligence forward. It is the quintessence of ‘those who don’t know their history are doomed to repeat it’. The difference within this scientific equivalent is there is no history to repeat when it comes to technology. And so it is from the humanitarian angle AI is considered. The ability to do great things with technology is writ in books and screenplays, and so are its dangers. While it isn’t likely an overabundance of ‘Mr. Smiths’ will fill our world, it is important we continue to break out of the siloes of science versus social sciences. If AI is to help humanity move forward, it’s important to ensure humanity plays a role in teaching our machine learning systems how different we are from each other and to consider the whole person, not just their exoskeleton. If you’re interested in the Data Sciences, Data and Technology, Machine Learning, or Robotics just to name a few, Harnham may have a role for you. Check out our current vacancies or contact one of our expert consultants to learn more.  For our West Coast Team, contact us at (415) 614 - 4999 or send an email to sanfraninfo@harnham.com.  For our Mid-West and East Coast teams contact us at (212) 796-6070 or send an email to newyorkinfo@harnham.com.  

Puzzle and Problem-Solvers: Software Engineers Drive Business

Software. It’s the drivers to your printer. It’s the word processor on your PC. And it’s the concept behind your productivity tools, your CRM systems, and your social media programs. Software engineers are to software what Data Engineers are to Data.  Software Engineers are the creators, builders, and maintainers of software systems and programs, so business runs smoothly. Now, that the majority of businesses have shifted online, it’s more important than ever to keep things running smoothly. These engineers must take into account not only what businesses might need to run, but also the limitations of the program. It’s a balancing act of software, hardware, limitations, and possibilities. If you took apart watches as a kid to see how they worked, Software Engineering might be for you. Are you a problem solver? Do you love putting the pieces of a puzzle together whether it’s on a board or in a crossword? Software Engineering might be for you. What Kind of Software Engineer are You? While there are a variety of roles to consider, below are some of the more popular paths taken. So, let’s say you want to build computer applications that affect what the end user sees. If you know programming languages such as Python and Java, and understand the mechanics of how to make a program work, then you may fit the classic example of a Software Engineer. If you’re more interested in the focus of robotics or automation, you may want to consider a role in Embedded Systems. You’ll still be designing, developing, and maintaining but your projects will be hardware and software used for a specific task.   Want to keep information secure? You may lean toward Security Engineer. In this role, you’ll ensure there are no security flaws. How? By operating as a ‘white-hat’ ethical hacker to attempt breaking into existing systems to identify threats. Technical Skills are Essential. Soft Skills are Important.  For anyone in the Data professions, technical skills are paramount. This not only gets your ‘foot in the door’, but ensures you know the basics. And for those who’ve been in the game a bit longer, also gives businesses confidence you can meet any challenges which may come up. Technical skills for Software Engineers include knowing programming languages like C++, Python, Java, and others like them. In this role, you’ll need to understand development processes as well as additional technical concepts. Technical skills are a standard requirement. And as important as it is to have a good portfolio and experience, you’ll want to show the business, you have the technical know-how to take on anything which may come your way. Now that cross-functional teams across departments are regular occurrences and C-suite executives are in the know, soft skills are just as important as technical skills. What are Soft Skills? In a nutshell, soft skills are communication skills. In the past, Data professionals may have been siloed away from other teams, and a liaison of sorts might have translated Data information into actionable insights. Now businesses and professionals have found it’s much more efficient to have the Engineer speak directly to their team, their leadership, or stakeholders. So, it’s imperative your soft skills are on par with your technical skills. Scope of Work for a Software Engineer According to the Bureau of Labor Statistics, Software Engineer employment growth is expected to grow 21 percent by 2028. Now that we’re working, studying, and socializing online more than ever, is it any wonder? Add to this the changing needs of organizations as they shift their practices into the cloud, and it’s more important than ever to have professionals who can design and maintain software to meet the needs of an organization. Whichever avenue you choose, whichever business you join or career path you follow, the full scope of work will be broad. You could be in charge of creating, developing, and maintaining a full product or just a single component of an app. Regardless of your scope of work, though, you’ll most likely be working with developers, cross-departmental staff, executives, clients, and stakeholders to mold, shape, and fulfill a design envisioned for their product. If you’re interested in the Data Science, Data Technology, Machine Learning, or Software Engineering, Harnham may have a role for you. Check out our current vacancies or contact one of our expert consultants to learn more.  For our West Coast Team, contact us at (415) 614 - 4999 or send an email to sanfraninfo@harnham.com.  For our Mid-West and East Coast teams contact us at (212) 796-6070 or send an email to newyorkinfo@harnham.com.  

Robot Companions in Senior Care Combine Computer Vision and AI

It’s 2021. Autonomous cars are no longer on the horizon. They’re here and being tested. Most businesses have shifted to a fully remote workforce or offer a hybrid option. And social activities have been redefined.  As social creatures, we humans crave attention. But does it matter from whom the attention comes?  Social distancing is a gold standard these days to keep the pandemic at bay as best we can. But what happens when the pandemic solutions affect one of the fastest growing demographics in the US?  Many active seniors have embraced the online – FaceTime and Zoom calls with friends and family, online classes for new experiences, and interactive activities to keep minds sharp. For those seniors in eldercare and assisted living, interactive has gotten a shape. Enter the robot. With Deep Learning, Computer Vision is able to enhance what a robot may see or what we see when we look into a robot’s LED face. We’ve worked hard to emulate the human experience in a machine, and have begun putting together instead a machine with human-like experience.  Unsurprisingly, robotics have become part of a variety of industries from manufacturing to construction to…eldercare? Socially Adept Robotic Companions in Senior Living Scenarios Since Jane Jetson ordered Rosey from U-Maid, we’ve wondered and worried about the roles robots might play in our lives. As we remain socially distanced and families and friends make contact through videoconferencing to those in assisted and senior living facilities, we’ve uncovered a new shortage of skilled workers. This time it’s those in the healthcare industries. Particularly those who care for the elderly. There is an ever-widening gap between healthcare workers and those who need them. In less than 10 years, there will be a shortfall of over 150,000 care workers in the US. In twenty years, that shortfall is expected to double. Recently, Robotic Researchers, Roboticists, and Data Scientists have been putting together plans for a robot much like Rosey, the maid was to Jane Jetson. Though expecting residents to only need or want help in things like delivery or picking up and delivering items, it revealed instead a desire for social interaction.  A prototype robot offers assistance from delivery to picking up items to karaoke and bingo activities. Add in a video-conferencing screen for interacting not only with friends and family members who are unable to visit, but also telehealth services with their doctor, or interacting with staff members who may not be nearby. Ways We’re Using Robots to Heal When we teach Artificial Intelligent beings and incorporate Machine Learning into our robots, we’re creating opportunities to heal. Already in use in healthcare from exoskeletons to assist stroke victims to Augmented Reality surgical practice, and real-life robotic assists in surgery, we’re able to help individuals heal physically. For many, the social isolation in eldercare homes can lead to depression and loneliness. But when someone, or rather, some thing is able to interact with them, some find a unique companion. For individuals who have difficulty connecting with people or those suffering dementia, it can be frustrating to not be able to communicate. But the role of robot in our lives just may bring a smile, a story, or a comfort. But robots aren’t just human-size companions. Some robotic companions are in the shape of pets. For those suffering from dementia, a robotic pet offers companionship and a less stressful alternative to live pets. There’s no need to worry about feeding Fido or Fluffy. These robotic pets love to be petted, but they don’t bark or meow, they don’t need to be let out, and bring to their caregivers a sense of purpose.  If you’re interested in the Data Sciences, Computer Vision, Machine Learning, or Robotics just to name a few, Harnham may have a role for you. Check out our current vacancies or contact one of our expert consultants to learn more.  For our West Coast Team, contact us at (415) 614 - 4999 or send an email to sanfraninfo@harnham.com.  For our Mid-West and East Coast teams contact us at (212) 796-6070 or send an email to newyorkinfo@harnham.com.  

Modelling the Mind with Computational Biology

Since Dolly the sheep was first cloned, humans have had a love-hate relationship with machines. Ok, maybe even before we asked a machine to make a living thing. In a variety of industries, machine learning systems, AI, and robotics are taking on the routine, mundane tasks once reserved for humans. But they’re doing this not to take away from humans, but to give them an opportunity to operate at a higher, creative level.  So, when you’re modelling the mind using Machine Learning and Computational Data in Neuroscience for mind blowing breakthroughs, we sit up and pay attention.  When it comes neuroscience, the benefits far outweigh the pitfalls. Just ask the researchers in China, who’ve developed a way to spot whether or not a child has autism from imaging the back of their eye. Other neurological orders such as dementia and Alzheimer’s falls under the computational neuroscience spectrum as well. From the 1970s to today, computational biology, using analytical, mathematical modelling, and simulation techniques to study behavioral and biological systems has evolved into a variety of subgenres. And it's within these subgenres we get a sneak peek into the mind of man that creates computers that can understand the mind of man. Can you wrap your head around it? Engineering the Mind – Mathematical Relationships The Life Sciences, Biostatistics, and Computational Biology all play a role in physical and mental health care. In seeking to understand the makings of the human mind, to study its syntactic rules, and to help explain how we think, human and machine have come together again. This time in the form of Computational Psychiatry. It’s here we realize our computational theories have often mirrored what we hoped to accomplish in building computers that could think with reason and logic. By understanding how we think, how the brain performs, and how it solves problems, can also help us to identify what we see as abnormalities of the mind – autism, schizophrenia, Alzheimer’s, dementia, and Parkinson’s disease just to name a few. At its heart, the fundamental message is that the brain’s way solving of inferred problems can be useful in determining hypotheses around neurological disorders.  Even within these subgenres there are varying degrees of theoretical concepts and with the data Computational Biologists and Computational Psychiatrists are able to conduct to navigate the complex inner workings of the brain. But much like the gathering, collecting, and analyzing of the data for the pandemic, the same can be done for in the mental health arena. Not the least of these theorems newly determined comes from a new theoretical model in the journal Medical Hypotheses. In it, T.A. Meridian McDonald, PhD, a research instructor in Neurology at Vanderbilt University Medical Center describes the positive traits of autism.  These positive traits she puts forth include but are not limited to increased attention, increased memory, increase differences in sensory and perception.  Building Computational Relationships Building relationships between neurobiology, environment, and mental signals in computational terms provides a cognitive model to understand the current state of one’s environment. It’s this building of relationships upon which human minds and the inner workings of the machine come together for the common good. There are positives in the negative. Mindset shifts aren’t just for learning how to work online or be more mindful, but are how best to present, and put your best foot forward. If you’re interested in Life Science Analytics, Computational Biology, Decision Science, Machine Learning, or Robotics just to name a few, Harnham may have a role for you. Check out our current vacancies or contact one of our expert consultants to learn more.    For our West Coast Team, contact us at (415) 614 - 4999 or send an email to sanfraninfo@harnham.com.  For our Mid-West and East Coast teams contact us at (212) 796-6070 or send an email to newyorkinfo@harnham.com.  

Recently Viewed jobs