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.