*Note: This blog post was written in draft form a few days after the event described.*

On September 24

^{th}, 2014, I heard of my sister's death from cancer while I was walking to a park (I was informed by a telephone call). I continued my walk to the park and reached it within 30 minutes. When I entered the park, a very strange thing happened. Leaf after leaf started to fall from the trees. One could describe it as a leaf shower. In the sixty seconds it took me to walk from the entrance of the park to a pond, it seemed like at least 100 leaves fell (it could easily have been 150 or 200).*The park where the "leaf shower" occurred*

Such an event would be
less surprising if it were in October (when most of the leaves fall in
New York City), or if it were a very windy day. But it was neither. The
average wind speed in New York City at the time was 11 miles per hour, and
the location was far away from any place where urban “wind funnel”
effects arise. The date was only September 24

^{th}, when relatively few leaves fall in New York City (peak foliage in New York City isn't until about October 17th). There was no noticeable gust of wind, just a gentle breeze.
I immediately sensed that
this event was very, very improbable. The odds against such a spike
in leaf falls (during a sixty second period) seemed astronomical.
For comparison, when I returned to the park on September 28

^{th}, I looked around for three minutes at the same spot where this “leaf shower” had occurred, counting the leaves that fell – but saw only one leaf fall.
But just how unlikely was
this “leaf shower” I saw on the day of my sister's death?
Calculating that turned out to be an interesting exercise in
probability calculation.

**The Probability Calculation Inputs**

Later I tried to do some
calculations to calculate the likelihood of such a “leaf shower,”
but it seemed extremely hard. Finally I figured out a technique for
making a probability estimate. It involved using an online “Binomial
probability calculator.” I used the calculator at vassarstats.com
for this purpose. Part of its main screen is below.

Like all such calculators
of probability, this calculator requires that you know three things:

(1) the number of trials
or tries (represented in the screen above as n);

(2) the number of
successful outcomes (represented in the screen above as k);

(3) the probability of
success on any one trial or try (represented in the screen above as
p).

But how could I make my
observations fit this calculator? I thought of a way. I decided to
invent the concept of a “branch minute,” and to use that as the
“trial” that the calculator would deal with. By a “branch
minute” I meant a particular minute of time during which one
particular tree branch might or might not shed a leaf. Under this
model a “successful trial” would be a case of a branch shedding a
leaf during a particular minute. For the “number of trials” I
would put the total number of nearby tree branches within my view
while I witnessed this “leaf shower.” Since my walk down this
pathway had only lasted one minute, the total number of “branch
minutes” would just be equal to the total number of branches near
the walkway. I estimated this as 300 (although I easily could have
used a number half as much).

For the “number of
successes” I would simply list the number of leaves I had seen
fall, which I estimated as being at least 100. But what about the
“probability of success on a single trial”? That fraction would
need to be the chance of one branch at this park shedding a leaf
during a particular minute on this not-very-windy September day. I
used the number of leaves on the walkway to help estimate
this number. Given the fact that there were relatively few leaves on
the walkway, the fact that it was only September 24 (about a month
before peak foliage in New York City), and the fact that the wind
speed was low, I estimated the “probability of success on a single
trial” as only 2%. This means that if you were to be staring at a
particular branch on a tree on this September 24th date, you would
have to wait an average of about 50 minutes before you would see a
leaf fall from that branch.

This 2% estimate seems
generous, and the actual likelihood may have been much less. I may
note that four days later I walked to the same place, and looked around for
three minutes, waiting for a leaf to fall. I saw only one leaf
falling. Judging from that observation, and using the same estimate of 300 nearby tree branches, I would have concluded that
the probability per “branch minute” of a leaf falling was only
about .1% (only a twentieth of 2%), because such a probability would
result in only one leaf falling along the walkway every three
minutes.

**The Output Probability Calculations**

So after putting in these
inputs, corresponding to the estimates above, I pressed the Calculate button. Here was the result that I
got.

The calculator indicated
that there was less than 1 chance in a trillion that this number of
successes (which in this case is the number of leaf falls I saw
during that minute) would have occurred. That is a probability less
than the chance of you meeting a stranger and correctly guessing on
the first try both his social security number and his birthday. The
calculation confirms my original thought that what I saw at the park
was fantastically improbable.

But what if I
overestimated the number of leaves that fell, would that make a big
difference? It turns out that using this calculator I would have get
the same probability (of less than 1 in a trillion) even
if the number of successes (the number of observed leaves falling)
was only 35. So even if only 35 leaves had fallen during the minute
of my walk to the pond, the chance of that happening would only have
been less than 1 in a trillion. This shows that the actual
probability of getting 100 successes out of 300 trials with a chance
of success per trial of .02 (2%) is actually something much, much less
than 1 in a trillion – perhaps as low as 1 in a quadrillion or 1 in
a quintillion.

**The Possible Significance**

Why is this possibly
significant? Because the odds of this “leaf shower” event
occurring naturally seem so incredibly small (much smaller than the
chance of you winning 100 million dollars in the lottery tomorrow),
there is an almost irresistible temptation to interpret it as a
possible sign from the Great Beyond, particularly given the fact that
it occurred within an hour after I learned of my sister's death.
It's not all that farfetched as it may seem. Imagine that there is
some spiritual world that somehow overlaps our physical world. It
might be possible for there to be a little energy interaction between
the two, of a type that might result in a wildly improbable falling
of leaves.

I know of one person who
saw a rainbow on the day of her mother's burial. She interpreted that
as a sure-fire sign from heaven. But seeing a rainbow on a particular
day is not terribly improbable. Given the number of rainbows I have
seen in my life, I roughly estimate that your chance of seeing a
rainbow on the day of a funeral or burial is about 1 in 5000. But
the probability reached through my calculation is vastly smaller.
Seeing a rare rainbow in the sky is chickenfeed compared to seeing
something as improbable as what I saw.

Humans are very bad at
comparing the probabilities of different events. I can imagine having
the following conversation:

**Me:**It was eerie. On the day my relative died on September 24

^{th}, I saw a shower of at least 100 leaves in a single minute, on a day with little wind.

**Jane:**Oh, that's

*nothing*. On the day of my mother's burial, I saw a rainbow.

But Jane's comment is
wildly off base. The probability of the first event is perhaps
billions of times smaller than the second event. So a more
appropriate conversation to have is one like the one below.

**Jane:**It was eerie. On the day of my mother's burial, I saw a rainbow.

**Me:**Oh, that's

*nothing.*On the day my sister died on September 24

^{th}, I saw a shower of at least 100 leaves in a single minute, on a day with little wind.

Some spiritual people say
there is no such thing as a coincidence. I don't agree. I think there
are plenty of “100 to 1 shot” coincidences and “1000 to 1 shot”
coincidences that happen all the time, along with rare “100,000 to
1 shot” coincidences you might see a few times in a lifetime. But
when you start considering a coincidence with a probability of much less than 1 in a trillion, that is when it
becomes very, very hard to believe that mere coincidence is involved.