Miracles No More?!

When we see or hear about an event that we did not expect it to happen, we react with surprise.  We describe them as rare, unusual, improbable, coincidence, unbelievable, and sometimes outright miracles.  Have you ever felt that these supposedly rare events seem to happen much more often than you expected?  I submit that many of us feel the same.  Further, throughout the human history, people have been seeking explanations why these improbably events happen.

One possible explanation is that many of such events indeed shouldn’t have happened.  Therefore, since they did, they must be caused by the power of some unknown non-human entities or forces.   Another possibility is that our expectation was too low.  We probably have failed to account for some factors and thus significantly underestimated the possibility for those events to occur.  That is, they are not rare events to begin with.  Or it is simply that we haven’t acquired sufficient knowledge and tools to provide adequate explanations.

To address this issue is by no means easy.  Fortunately, with the advances in philosophy, sciences and mathematics, we have already come a long way in clarifying and quantifying the issue rationally and logically.  In fact, there are decent tools now for us to tackle the question if the occurrence of certain events is likely to be truly unusual.

But the harder problem is actually in convincing those who thought an event is a miracle when it is not in the first place.  Part of the challenge is to show the reasoning to laymen who are not familiar with the language and tools of mathematics and statistics.  The landscape has changed earlier this year when David Hand, professor emeritus of Imperial College London and a former President of the (British) Royal Statistics Society, published his beautiful little book The Improbability Principle – why coincidences, miracles, and rare events happen every day. 

As the title indicates, Professor Hand showed us eloquently with many real-life examples and stories how easy and often we get impressed with events/news that we thought should not have happened but did.  He identifies the main culprit what he called Improbability Principle – a collection of strands/laws, each of which is sufficient to produce outcomes that appear to be highly improbable to the uninitiated.  What makes the matter much worse and so difficult to discern is that these strands of Improbability Principle can “intertwine, braiding together and amplifying each other, to form a rope connecting events, incidents and outcomes.”

To give you a sneak preview, in Chapter 4 of the book, Professor Hand brings forward the Law of Inevitability which says “even if each of the possible outcomes has a tiny probability of occurring, it’s certain that one of them will.”  One example he gives is the stock tipster scam whereby the perpetrator sends letters to say, 1024 individuals and claims falsely to have the ability to predict the daily ups and downs of stock market.  Most of us would dismiss such a claim instinctively.  But the scammers are pretty smart people too and have their ways to combat the skepticism, making use of the law of inevitability!   The perpetrator would send 1024 distinct predictions of market movements of 10 future business days, one for each of these 1024 individuals.   If you and I are making a random guess, we would have less than 1 in 1,000 chances to get it right.  But guess what?  Since there are only 1024 possibly distinct outcomes of up-down movement over any 10 business days, one of these 1024 individuals is guaranteed to have received from the scammer the correct 10-day long pattern.  Such an incredibly impressive achievement (to the one who received the correct prediction) could be sufficient for the perpetrator to profit from the scam follows.   

In Chapter 5 of the book, Professor Hand recalls a number of interesting real stories, ranging from someone had won multiple lotteries over a short period of time to a woman saw a book in a used-book store that she used to own many years earlier.  You and I would probably react to such stories with “Wow, what an amazing coincidence!” and soon forgot about it.  Then we find ourselves react similarly soon enough with yet another such a story.  To show us why such stories seem to take place so frequently, Professor Hand introduces the Law of Truly Large Numbers that says “with a large enough number of opportunities, any outrageous thing is likely to happen.”

Take the lottery story as an example.  Most of us understand that the chance for a randomly chosen player to win a huge jackpot can be astronomically small (by design).  Further, most of us also understand that the chance of an individual would win two large jackpots of independent lotteries is exponentially smaller – the odds to win both would be a trillion to 1 if winning one is a million to 1.   But then, should we be surprised upon hearing the news that someone had won two lotteries somewhere?  It turns out that we really shouldn’t be surprised.  Part of our problem is that we were looking at the wrong measures.  While the chance of winning a large jackpot of a lottery is indeed tiny, what the news and we were actually talking about was the (expected) number of occurrence of such an event over an unspecified period of time – or how many winners would you expect to have such an unbelievable luck of winning two jackpots.   Were you ever surprised when someone did win the jackpot of a lottery?  Once you realize how many lotteries there are and how many people are playing multiple lotteries and bought many tickets regularly, why should we be surprised if once in a while, one of them won two lotteries somewhere over a long period?  The lesson is that although the probability for an event to occur is tiny, we tend to forget the number of attempts is can be much larger than we expected and consequently, the event is almost certain to happen in due time.

These two are just tips of the icebergs.   Human beings do have at least a few more blind spots that can mislead us as well.  In chapter 6, Professor Hand notes the Law of Selection that says “you can make probabilities as high as you like if you choose after the event”.  While I trust none of us would cheat by drawing the target after shooting the arrows, we do commit, from time to time, unconsciously the act of fitting questions to answers and identifying causes to result after the event has taken place.  When was the last time you heard someone attributes a good thing happened to him/her (after it happened) due to his/her prayer made earlier?  What he/she might have neglected was the other nine hundred ninety nine times when nothing memorable happened!  No one was harmed in these type of stories of course.  But what if it is a scrupulous investment advisor who is telling you examples of selected clients who had made insane profit?  Would you be so impressed and invest your life time savings to his scheme?  By the way, this is closely related to the so-called Confirmation bias that we tend to count and remember only those events that you want to include for whatever the reason.

If all these were not enough, we definitely are not helping ourselves by getting little too relaxed or sloppy.  Complementing the Look-elsewhere Effect, Professor Hand’s law of near enough says that events which are sufficiently similar are regarded as identical.  With this powerful law, you will see that we can ensure the “connection” we want to draw can be true.  

Let me show you with the following example.  Let us say you try to initiate a conversation and “connect” with a girl in a bar.  First, the Look-elsewhere effect expands the space of search – if she did not come from your town, ask if she is coming from the same county or state.   If that is not the case, the law of near enough will likely do the job – you can ask if she knows someone from the state you come from.  You will be impressed how effective this trick works.   I used to brag about I can find some connection quickly with any stranger from Taiwan – a place of 26 million people.   That was before I learned that someone has already formalized a theory of Six Degrees of Separation that everyone and everything in the world can be connected within six “connections”!   Interestingly, most of us accept matching 5 numbers is not worth much in a lottery where the jackpot requires the matching of all 6 numbers.  The trap is, while 5 deceptively close to 6, the cases of five matches is so incredibly larger than that of matching 6!

There are a couple more laws the book discusses.  You can read them at your own leisure and reach your own conclusion.  Even if you don’t agree with some of points made in the book, you should at least be aware that many legit as well as scrupulous people/organizations do use these laws to their advantage to try to get your money.  If you do find the analysis interesting and helpful.  Here is an exercise for you.  When you open a fortune cookie the next time, tell us if it is relevant and applicable to you and if any of these laws are at work.  

Sorry, the world is more boring than you and I thought.  Believe it or not, there simply aren’t as many improbable events and miracles.   Talk to you soon!