When Summing the Parts Doesn't Add Up

We work with companies to understand the value of discovering and influencing the uncertainties associated with their strategic and project decisions.  Usually there are few key uncertainties whose range swing the value of the decision by a large margin.  Understanding those drivers, their impacts, and the result of influencing their outcome is what we call the Insights to Success.

However, sometimes the number of uncertainties that must “go right” can be so large that even a small degree of risk for each one makes the project appear hopeless.

Consider this problem – for the Apollo spacecraft to successfully go to the moon and back, literally thousands of things had to go right.  If any single item failed, then the launch, landing, or return would fail and a disaster would have ensued.  The problem comes when the many uncertainties are totaled into an outcome.  Ten uncertainties, each with a 99% chance of success, results in a 90% chance of overall success (0.99¹⁰).  One hundred uncertainties, each with the same chance of success results in only a 37% chance of overall success.  Increase the number of uncertainties to one thousand, and the flight is pretty much guaranteed to fail.

So we argue that the chance of success for each part is actually greater than 99%.  Let’s assume the chance of success for each part is actually 99.999%.  The same problem occurs as the number of parts increase.  When we get to 100,000 parts with a 99.999% chance of success for each, the total chance of success is still almost nonexistent.

This problem also occurs in business uncertainty assessments.  When the business case is broken into a large number of uncertainties, the overall project may start to appear to have a high degree of failure.

I call this the Uⁿ or “U to the n” problem.  The uncertainty, raised to the number of uncertainties, starts to make the resulting value small and can quickly disconnect the problem’s relevance from the real world.

This is, in part, because we humans have a difficult time grasping very large or very small numbers.  Our reference points are often influenced by our macro experiences.  We know we are late for the airport about 1 in 100 times, and since that doesn’t seem like very often, a 1% failure “feels” like a low probability.  Trying to get our minds around a 0.0000001 chance of failure is almost impossible for most of us to grasp.

So how do we assess the very small probabilities of failure?  There are two approaches I’ve seen which works.  With the first approach is to aggregate the uncertainties into groups or systems, and then assess the probability of the system succeeding or failing.  Compare this assessment to the uncertainty obtained from calculating the individual uncertainties and see if the numbers are close.  If they are not, perform a logic comparison of the system assessed uncertainty to the calculated individual uncertainty effect.  Why are they different?  Which individual uncertainties are driving the number to be off when compared to the system assessment?  Also, it’s invalid to state at this point that one number or the other is “right”.  It’s more important to resolve the differences to get to the correct assessments.

The second approach uses comparative assessments.  Compare the likelihood of an individual assessment against one or more uncertainties with a known track record. For example, rank the following in the order of likelihood:

·         A bolt shearing off from the reactor housing

·         The fuel source becoming contaminated

·         Being more than an hour late for work

·         Finding a wrench forgotten in the fuselage

·         Having your air conditioning fail in your house, car, and office at the same time

This approach helps you calibrate your assessments into the right range of numbers before assigning a set of probabilities.

Sometimes, both approaches are necessary to fully grasp and analyze a set of uncertain events.  By testing both approaches you will be applying both the intuitive and reason approaches to the problem.  Some team members will be better at one than the other, but by testing both, you will have a more robust and valid assessment.

Of course, the prevention of failure is often the driver for redundant systems.  For each backup, the probability of failure is decreased the same amount as lowering the n in the Uⁿ by one.  That is, it’s a logarithmic benefit and indicates the mathematical basis for the the high value of applying redundant safety systems.  But that’s a topic for another time.

Defining "Success"

This blog addresses the theme of understanding the Insights associated with achieving success.  If you’re reading this from our website at KCarpenterAssociates.com, you will find a lot of information about how we help companies achieve success as measured by financial return.  We have a rich history of helping companies understand the value of their projects and decisions across a broad set of uncertainties, and forming the Insights to Success that enhance their projects and strategies far above the initial base projections.

Our approach works for individual success as well. I have guided many developing employees in understanding what success means for them, and how to see the uncertainty in their future as an opportunity to enhance their level of accomplishments and their chances of succeeding.

However, we are all subject to the biases of our culture and many people simply assume that individual success is measured by financial accomplishment.

So let me tell you a personal story about just how wrong this default approach can shade our thinking.

I grew up in the Midwest.  Both of my parents were educators during much of their professional careers.  My Mom held a BS in Biology and a Masters in English.  She taught students from Kindergarten through Middle School in Missouri and then continued teaching adults in Saudi Arabia.  My Dad held a PhD in Chemistry and was a Professor at the University I attended in Northwest Missouri (Go Bearcats!), before returning to industry with Aramco.

I hold a BS in Chemistry with an MBA.  Education is important in my family and I naturally assumed it is a pre-requisite for a successful life.  If by success, we mean financial opportunities, then this is the normally accepted path for most of America to “get ahead” and “succeed”.

However, when we define success as Happiness, our paradigm quickly changes.  The press is full of stories about “rich but miserable”, as well as “less financially accomplished but happy” people. Happiness cannot be measured by a financial metric alone. Happiness is a function of many diverse drivers that most people have internalized but may have difficulty articulating. 

My son however, was clear about what would make him happy when he discussed his plans with me one day.  It was near his high school graduation date when he said “Dad, I know you want me to go to college, but I just don’t enjoy learning through a classroom. I like hands-on learning and I want to attend technical school to learn how to work on engines.”  Ross is a quiet kid, so it took a lot of courage to discuss this with me, and to have the strength of his convictions to talk about what would make him happy.  He also didn’t want to remain in our big city of Houston, and was interested in moving to a smaller town, back in Missouri.

I confess that his news was not met with the warmest reaction from me, and I could have been far more encouraging and supportive.  But we talked about it and he convinced me that he would be far happier with his approach than the plan I had envisioned for him.

Ross enrolled in the Universal Technical Institute, and while he was an average student in High School, became a high performer at the technical school, graduating this spring from the advanced diesel class with honors.  While attending school 30 hours a week for the last two years, he also held a full-time job at an auto parts store, working his tail off seven days a week and enjoying every minute of it. 

He’s just returned from Springfield Missouri where he’s secured a good job doing what he loves.

Ross has already achieved more success by age 20 than many people struggle to achieve through their lives.  I know he has the skills, drive, and character to carry him to further accomplishments, and will be a big success in life, regardless of how the world chooses to measure a winner.

When we redefine success as Happiness, we begin to reconsider our objectives – those goals and achievements by which we measure our lives.  For some, it may still be driven largely by financial accomplishments.  For others, it’s making a significant contribution to society.  For some, it’s working on challenging problems.  And for others, it may be as simple as doing what you love.

Ross was fortunate.  He found his definition of success at a young age and will have a rich and rewarding life as he pursues his dreams.

For most of us, it seems to take a bit longer.

And on a personal note - Ross, let me add – you and your sister are the most significant achievements of my life.

Love, Dad

When the “Expected Value”, isn’t

A core concept in uncertainty-based decision making is the “Expected Value”.  Simply stated, if an uncertain trial is repeated many, many times, the Expected Value is the average result you would experience across all of the trials.

A simple illustration can be constructed using a coin.  Image a game where heads wins the player $100 and tails returns $0.  For each flip, the player either gets $100 or $0.  If the player repeats the game a hundred times or so, he or she would receive on average $50 for each trial.  The Expected Value for this game is then $50.  But what’s the chance of the player winning $50 playing the game once?  0% of course.  They will always win $100 or $0 on a given flip, but NEVER $50.

Crystal clear right?  Couldn’t be simpler you say.

But here’s where the problem arises.  For complex projects, businesses often construct uncertainty-based business cases.  We consider many uncertainties and model the range of their values, resulting in a ranged value for the business case.  We can see the distribution of possible outcomes and calculate the average of their values.  Invariably we label this the “Expected Value”.  

What’s the chance of actually getting this outcome?  Once again, it’s practically 0%.  It is but one point on the curve of possible outcomes.  However, when we place a grand label on this data point, suddenly everyone forgets the distribution of possibilities, and expects this value to be achieved.  For most assessments, if the Expected Value is about in the middle of the distribution, there is roughly a 50% chance you won’t get that good of a result.

Uncertainty-based business analysis is more about exploring the distribution of outcomes and their drivers, and taking action to improve the results based on these Insights, and far less about a single point on the outcome distribution. Comparing only the Expected Values of project alternatives is not a valid approach to selecting which project to promote, and if the Expected Values are close, they should be ignored completely.  Examining the breadth (or risk) of the outcome distributions, their key drivers, and the probability to experience disaster or wild success is far more insightful and will make for better decisions, stronger projects, and more successful companies. 

If only we had called the point something other than “Expected Value” all of this confusion could have been prevented.  I suppose we could have simply called it the “Mean Value”, but then we probably would have to devise an additional analysis to find the “Nice Value”.

A Tax on the Mathematically Challenged

Statistical logic is often forgotten in the face of emotional decisions.  This is one reason I do not play the lottery.  Sure, over the years, I’ve been given scratch-off game cards for birthday presents.  I will admit that I’ve scratched off the silver paint with all the excitement of an 8-year-old at Christmas. And usually, I have the same disappointment of the 8-year-old when he realizes the brightly decorated package contains socks.

In general, I consider the lottery as a Tax on the Mathematically Challenged.  But even the smartest people continue to play the game as they let their emotions (and the hype) drive their decisions.

Consider a 6 number lottery.  Each number can be from 1 to 50.  If no number can be repeated, the chance of guessing the correct 6 numbers in no particular sequence is about 1 in 16 Million.

I have a friend who is a Really Smart Guy.  He has an advanced degree and understands statistics quite well.  He was explaining to me with great excitement how he only plays this type of lottery when the payout exceeds the risk.  That is, when the prize is greater than $16 Million, he puts in his dollar as he calculates that the large payout multiplied times the small chance of winning exceeds his $1 investment.

I may have ruined his day when I pointed out that more than one person could win the lottery, and reduce the payout, destroying his statistical logic.  In fact, as the payout increases, more and more people play the lottery, increasing the chance of having to share the winnings and reduce the amount he might win.

Perhaps I should have remained silent and let him have his fun.

You will find many websites that explain how you’re less likely to win the lottery than get hit by lightening, crash in a plane, or have a car wreck, and that you would be better off putting your investment into insurance against those types of events.

However, if after all of this information, you are still set on playing the lottery anyway, there is a way to reduce the chance of sharing the prize and increase your chance of a winning payout by applying a little game theory. 

As all numbers are equally likely to be selected, select numbers that other players are less likely to pick.  People are often superstitious in their selection of lottery numbers, picking dates that are associated with birthdays, anniversaries or historical events.  This means they pick months (1-12) and days (1-31).  If you instead select all of your numbers to be in the range of 32 to 50, you are just as likely to win, but less likely to have to share the prize with others.  Unless of course if everyone reads this blog and all start to pick numbers in the 32 to 50 range!

When statistics go bad

Most of us understand the basics of statistics. Simple ideas, such as the concept of an Average, is taught with the very basics of elementary math.

Perhaps that's why it's so disturbing when we see them bantered about for evil purposes.

Last week, a televised news report was aired slamming the President on his energy policies and their direct contribution to the high price of gasoline. This reporting network is well known as being in support of conservatives and against the current administration.  However, regardless of your politics, they cannot be excused for their slanted ignorance when they stated - "Almost half of the consumers are paying more than the average price of gasoline".

Now, I'm sure many people didn't hear the subtlety of the comment. Some people may have been upset at how half of us are being subjected to a higher than average price.  What we should be upset about is how a news channel, that supposedly hires college graduates, can let such a comment make the air.  Then again, perhaps they hire the lowest 10% of the class.  I understand they make up almost 1 in 10 of the graduates.

Next week we'll discuss how the lottery is a tax on the mathematically challenged.

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