Meetings

 

We’ll Meet Again

We'll meet again,
 Don't know where, don't know when,
But I know we'll meet again,
Some sunny day.

Math has something to add about this.

Meeting means that you are in the same place at the same time as another person.  That does not mean that the probability of your being in that same place at the same time as another is the same, but that your two probability distributions overlap.

Again depends on whether the event repeats. A normal random distribution is a logistics distribution, also known as the sech squared distribution because its Probability Density Function, PDF, uses the hyperbolic secant function,  1/(4s)*sech²((x-μ)/2s).  The mean location where you are is µ. The most probable location you will be is that location.  The odds of you being at location x is defined by the formula.  S is the range of the probabilities of your being at another location.  A standard normal distribution is one where there is a 50% probability of your being at the mean location, in which case s must be equal to 0.5.  If you and that other person share a very similar mean location, and you have met once, then there is a high probability that you will meet again, as shown in the graph on the left below.  But if your most probable locations,  are very different, i.e. you both are ordinarily not in the same place at the same time as shown in the graph on the right below, then the chances of your probability distributions overlapping in the first place was very small.  The “again” depends on the period in which these probability distributions repeat.

And this is where math comes in.  The most probable meeting, as you can imagine, is halfway between the most common locations, e.g. your homes.  This is true regardless of whether the other person is your neighbor or not.  This might be obvious from the graph on the right since the probability of a meeting is the product of the two Probability Density Functions, PDFs.  However, what might not be obvious, but it is also true, is that the probability of meeting at one person’s most common location, is the same as meeting at the other person’s most common location regardless of whether the other person is your neighbor or not. I.e. if the two PDFs are normally distributed, then the product of those PDFs is also normally distributed.  

But a hyperbolic secant function does NOT repeat in the real plane.  It has a period of 2πi, where i is the imaginary number, √-1.  This means that a random event only repeats in the imaginary plane.  This is unlike the conventional trigonometric function, sec(x), which repeats cyclically in the real plane with a period of 2π.  So if the meeting was truly a chance random encounter that had a very low probability of occurring in the first place, you may only meet again in an imaginary plane.  Random events do NOT repeat in the real plane.  A once in a lifetime event, will only occur once in your lifetime.  That “when” will occur again, but that may only be in your imagination.