Should Rare Events Surprise Us?

(Archive Question of the Week)

We often receive questions saying something like, “My wife has the same birthday as my son-in-law, whose dog has the same name as my wife’s brother, … ,” and asking us to calculate how likely it is that this would happen. Sometimes it is simply impossible to answer (without, say, looking up statistics on human and dog names); but even when an answer can be calculated, there is a fundamental fallacy in the question itself.

What’s wrong with the question

I often respond by showing whatever calculation I can do, together with a recommendation to read the following, where Doctor Vogler discusses the fallacy:

On Spins and Surprises

In roulette, what are the odds of a single number, in this case 15, hitting six times in nine spins?

I actually saw this happen and would like to know how rare an occurrence this was. I can figure the odds for one spin -- there are 38 possible outcomes on any spin of the roulette wheel, so you have a 1:38 chance of a single number hitting on one spin -- but don't know how to do this for multiple spins.

This is a fairly routine probability question, and Doctor Vogler answers it; but first, it is necessary to clarify what the event really is. The problem is that the event was identified after the fact, which skews the probabilities. And this was done because what happened seemed so surprising. But the reality is that surprising things (even extremely rare things) happen all the time. Doctor Vogler starts this way:

We get a lot of questions of this form. People come to us saying things like, "I just learned that four students in my class have the same birthday! What is the probability of that happening?" Or, "I was playing poker, and I got two straight flushes in a row! How rare is that?" Or, "I flipped a coin six times yesterday, and every time it came up heads! What are the odds of that?"

You should realize that rare events happen occasionally, and this causes people to ask how likely it was to happen. But they don't notice all of the many times that unlikely events *didn't* happen. So it's really not fair to say that you are defying probability by computing the odds after the fact.

For example, why is it that no one comes to me saying this?

"I was playing roulette the other day, and I played six times. The first time it got a 15. Then it got 7, then 19, 1, 5, and finally 27. What are the odds of that happening?"

This event is not at all surprising; but it is actually rarer than the event Ty is asking about! The difference is that somehow the same number coming up repeatedly seems more “special”. But if the question is really, how likely is it that such a “special” thing should happen, then maybe we should include other equally special things, so that the event Ty is asking about should be something more like this: “What are the odds that some one number will hit at least six times in nine spins?” And, since Ty was surely watching more than nine spins, perhaps the question is really, “What are the odds that some one number will hit at least six times in some nine consecutive spins out of the fifty I observed?” And even this omits many other events that would have been equally surprising, such as hitting the numbers 1 through 9 in order on the nine spins, or hitting the ages of my children in alphabetical order!

Doctor Vogler continues,

What are all of the outcomes that you might have seen that would have caused you to be surprised and ask what the probability is?

That's what I mean when I say that rare events happen all of the time. If you ask *before* spinning the wheel, what is the probability of getting 15 in six of the next nine spins, then my calculation would be valid, and of course you would lose your money if you got 23 six times or got the numbers from 1 through 9 in order. But if you see some event and then ask what its probability is, then in some sense the answer to your question is 1 (that is, 100%) because as you already saw, it happened and there is nothing that you can do to change that fact; the outcome is already determined.

Similarly, if you ask *before* spinning the wheel, what is the probability of getting the six numbers 15, 7, 19, 1, 5, and 27 in that order in the next six spins, then the probability would be 1/38^6, and you would be shocked out of your mind and suspect a magic trick if it actually happened. But if you saw those numbers in that order and *then* asked what the probability was, then someone telling you 1/38^6 would seem clearly wrong, because it wasn't that surprising an event.

Does that make sense? I just wanted to make sure that was clear before I did the calculation for you, because you might find the number smaller than you expect, and I hope you now understand why.

The calculation

With that out of the way, he does the calculation:

Okay, now let's suppose you are going to make a bet with someone, and the bet is this: I am going to spin the roulette wheel nine times. If it ends up as a 15 for exactly six of those nine times, then you win.

I will now calculate the probability of that event happening.

On each spin of the wheel, there are 38 equiprobable outcomes, which means that in nine spins, there are 38^9 equiprobable outcomes, each of which is an ordered list of 9 numbers. How many of those 38^9 outcomes will have 15 exactly six times? Well, that requires six of the nine numbers to be 15, and three of the nine to be *not* 15. There are 9-choose-6 ways to decide which of the nine are 15 and which are not, and no matter how you choose those, there are 37 ways to pick each of the three non-15's.

So the total number of outcomes that have 15 exactly six times is

   (9-choose-6) * 37^3 = 84 * 37^3
                       = 4254852.

So the probability of that happening is

          4254852/38^9 = 1063713/41304025315712
                       = 1/38830046.55928...
                       = 0.0000000257532526641...

Of course, if you want to change the bet to be "at least six" instead of "exactly six," then you should add

   (9-choose-7)*37^2 + (9-choose-8)*37 + (9-choose-9).

And if you want to say "some number" instead of "15," then you should multiply by 38. But this still won't include other rare events, like getting a 1 on every odd spin and a 2 on every even spin, or discovering that the dealer has the same birthday as you, or the same first name, or that this email arrives in your inbox at precisely the time (minutes and seconds) matching the birthday (day and month) of your great Aunt Silvia. But you can ask me those probabilities separately if you want, as long as you realize that I won't be accepting any bets after the outcome has already been determined.

Another example

Here is a typical question of this sort that was not archived:

What are the odds that two people getting married share the same birth dates as the groom's great-grandparents, and they are getting married 100 years later (not on the same date, close though, within 3 weeks)? Also, the bride is older than the groom, just like the great-grandparents. Thank you for your consideration. If you respond, I'd like to read the results at their wedding.

Doctor Vogler posted a memo to the rest of us, telling us that he had been about to answer this by simply referring to his answer above; but he thought it might deserve a more romantic (or at least sympathetic) answer! Here is what I wrote:

We get a lot of questions like this, and generally although it may be possible to answer what they are literally asking, it is not very meaningful. You may want to read this, which will explain why we don't really want to answer such questions:

  On Spins and Surprises
  http://mathforum.org/library/drmath/view/76807.html

In particular, the probability you are asking for would become much more likely if you included other similar cases, such as if it were her great-grandparents rather than his, or the dates could be swapped, or you included parents and grandparents as possible dates to match. Also, there is no way to calculate the probability of being married almost exactly 100 years apart; you'd need a statistical study to find that.

But you aren't looking for a technically perfect answer, just for something fun to say! So I'll start over.

Suppose you looked at lots of couples and found the groom's great-grandparents' birthdays. How common would it be that her birthday matches the great-grandmother's, and his matches the great-grandfather's? Well, given any two people, the probability that they have the same birthday is about 1/365, because the first person determines what the day has to be, and there are 365 possibilities for the second person. And when you want the probability that two independent events will BOTH occur, you multiply the probabilities. So the result is that the probability of both birthdates matching is

  1/365 * 1/365 = 1/133,225 = 0.0000075

This is 7.5 out of a million. I find that there are about 2.5 million weddings every year in the U.S.; so this can be expected to happen only about 19 times a year in this country, on average.

Unique? No. Special? Sure. But they already knew that, didn't they?

Now, it just happened that the day on which I wrote this was my own anniversary. What are the odds … ?

References

Here are a couple other discussions of this idea of common coincidences, that I have often referred to along with Doctor Vogler’s answer:

Understanding Uncertainty: Coincidences

Cut the Knot: Coincidence

 

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