## Tuesday, December 1, 2015

### An Immensely Improbable Event Shortly After Someone's Death

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

On September 24th, 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 24th, 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 28th,  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 24th, 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 24th, 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.