The faculty, staff, and student body at BPI often chatter about coincidences, and history offers plenty of coincidences. Like the story of Kevin Stephan, a Buffalo man who saved the life of a woman who had saved his life 9 years earlier. That’s too amazing to be mere coincidence, right? Well….

Actually it’s not all that amazing. While the probability of any single trained rescuer saving the life of someone who had previously saved his/her life is remote – on the order of one in millions – there are tens of millions of trained rescuers in the U.S. Even if you limit rescuer-rescuee pairs to those living in the same city, as Stephan and his rescuer-cum-rescuee did, that’s a whole lot of possible pairs.

A whole lot, as in … the probability of a story like Stephan’s happening somewhere in the U.S. in a given decade is about 1-in-3. No one thinks it astonishing that someone was born on a Monday – a 1-in-7 event – so why would a 1-in-3 event like Stephan’s amaze us? Yet on a TV show this week I heard explanations ranging from Jungian synchronicity to quantum entanglement.

It turns out we’re not very good at intuitively guessing how likely something is, or even at weighing known probabilities. Most poker players know a pair of Aces in the best starting hand in Texas Hold’Em, and that pocket Aces are about a 6:1 favorite to win over any other starting hand. Put another way, the probability that pocket Aces will lose to a random hand is the same as the probability of a person being born on a Monday: about 1-in-7.

Yet I can’t count how many times I’ve heard poker players use words like “unbelievable” after losing a hand with pocket Aces. I’ve even heard players claim cheating as the only explanation … which is like claiming the only explanations for being born on Monday are artificially induced labor or a scheduled Cesarean section.

Realworldia is a big place, and history is a long time. Lots of things happen, and there being only 365¼ days in an earth year, that means lots of things happened on any given date in history. Sometimes it’s not even mere coincidence. There were more recorded battles fought in late-spring and summer because that was the traditional military campaigning season: after crops were planted and before harvest time and the bad winter weather. Likewise for the founding of countries, a surprising number of which happened in late June and July.

I play with these coincidences for fun, but please don’t take them too seriously, nor think that I do. And be suspicious when someone says “That’s too amazing to be just a coincidence.” Coincidences happen … every day.

Coincidences do indeed happen every day. 🙂 And we’re wired to see meaning in them. Which raises a truly interesting question: How is it that we’re wired in such a way that our psychology flies in the face of statistics?

Or does statistics fly in the face of reality? Because honestly, when you come right down to it, we create reality inside our own heads. Don’t believe me? Look at the Tea Baggers.

Hugggs and good morning!

Our tendency to see patterns and infer cause-and-effect are very useful. There often are causal links, and many of the patterns we see are reliable enough to warrant our attention. That somewhat-horizontal line in the brush may be merely the wind moving through the grass … but it also may be a lion stalking the camp. Finding those patterns is useful enough that we do it even when the pattern is simply random noise, like seeing ‘faces’ in the wood grain on a board or door. Knowing which is which … we’re not so good at….

Of course then you also have to deal with the reality that if you flip a coin it will come up heads more often even if the probability is supposed to be 50-50. The more you flip the coin the closer they get to 50-50 but even with a million flips, heads comes up more often.

Coincidence or coin-incidence?

You sent me scurrying through the Googles on this one. Turns out you’re right, according to some mathematicians, but only if the coin is placed heads-up on the thumb before flipping. The physics are very complex – it’s a 31-page paper that will make your eyes glaze – but for a normal human flipped-off-the-thumb toss where the coin is placed heads-up to start and caught rather than allowed to drop to the floor or ground, the coin ends heads-up about 51% of the time.

However, this part isn’t quite true:

In fact you are more likely to confirm the 51% bias with a million flips. It’s not unusual to get several consecutive heads-up or tails-up flips, so in 10 flips you might get a 5-5 split … but 6-4 or 7-3 splits are not unlikely. In a million flips, the Strong Law of Large Numbers comes into play and you’re very likely to get an outcome set that matches the statistical probabilities: about 510,000 heads and about 490,000 tails.

Sorry to send you scurrying through the Googles. If I was a good fellow faculty member I would have linked to the Googling I had found to make my claim about it getting closer to 50-50. Of course, in a perfect world, I would have also

readthe text instead of just looking at the table and I would have discovered this factoid “I’ll illustrate below with made-up numbers. (I didn’t do a simulation.) “. Sheesh.I had a sense that it had something to do with the physics because the odds of one of two things happening if they were physically identical should be 50-50. A coin is not perfectly balanced.

Yes, the source you cited answered a slightly different question: why the difference between the number of heads-up and tails-up outcomes – if they were exactly 50-50 probabilities – will tend toward but rarely quite reach zero in a very large sample set.

With 1000 attempts and exactly 50-50 probabilities for each of two outcomes, your sample space – the total number of possible event sets – is 2^1000. That is a huge number, and surprisingly few of those event sets have exactly 500 of each outcome.

As for why a coin that begins heads-up on your thumb is slightly more likely to land heads-up in your palm, it’s not that the coin is unbalanced. There is a reason, but it has to do with very complex physics of angular motion.