## A few mathematical curiosities

As noted on my Probability treatments in [other books] page, there is a set of cute elementary math calculations or paradoxes found in many textbooks and popular science style books. Another interesting one, surprisingly less well-known, is the friendship paradox, which in informal language says "your friends have more friends than you do, on average". This is well explained in the Wikipedia link.

Here are some less well known examples.

### 1. Are you related to your ancestors?

This is phrased as a paradox; the actual question is whether you are genetically related to your ancestors. You have 23 chromosome pairs, so under the over-simplified story in which each of your chromosomes is a copy inherited from one parent, then you could be "related by descent" to at most 46 ancestors in any previous generation. Adding more detail makes the model more interesting: crossover implies that sometimes your chromosome is a splice of two parental chromosomes, thus arising from two different grandparents. A simple toy probability model from this 1983 Kevin Donnelly paper leads to the conclusion that you are "genetically related" to more than half of your 9'th generation ancestors, but less than half of your 10'th and subsequent generation ancestors. So pride in being descended from some famous person several centuries ago may be unjustified.

Imagine a hypothetical clock-like device which each night would increment by the probability that you will die the next day. If you are now 30 years old, this would now probably read around 2%. When someone dies, their "risk clock" will show some value, maybe 68% or 155%. But here is a remarkable fact. Everyone, regardless of their environment and choices, has the same probability distribution for the value at death: about a 50% chance to reach at least 70%, a 37% chance to reach at least 100%, a 13% chance to reach at least 200%.

Although this may suggest that by avoiding all voluntary risks you could live a long time, alas the hypothetical clock includes risks of dying from natural causes and involuntary risks.

Could this be more than hypothetical? In principle, in the near future you could have an AI (expert system) track all your activities and estimate the risks. Right now, the closest analog is the notion of your effective age based on lifestyle choices.

### 3. Gaming a prediction tournament

Our scoring a prediction tournament section gave the correct scoring method. One might think that the way to judge the accuracy of estimated probabilities is via
of the events estimated as having 60-70% chance, about 60-70% should actually occur
(and similarly for other ranges). This calibration property is certainly desirable, but using this as the only criterion is unwise. Suppose that currently less than 60% of your estimated-60-70%-chance events have actually occurred. Then next time there is an event which you honestly estimate as having more than 75% chance, you could announce your estimate as 60-70% chance. Via this strategy, with only the ability to estimate probabilities quite roughly, you can satisfy the calibration property. A major feature of the MSE scoring method is that such dishonest announcements are penalized.

### 4. The halftime price principle

In a sports match between two equally good teams, at the start the probability for the home team to win will be 0.5 (ignoring home field advantage). At halftime there will some (implied by gambling odds, say) probability for the home team winning. This probability varies from match to match, depending largely on the scoring in the first half of the match. Theory says its distribution should be approximately uniform over the whole interval [0, 1].

### Footnotes

The distribution in Example 2 is the Exponential(1) distribution. An implicit assumption is that there is no very likely probability of death on any single day.

The (simple) math for Example 4 can be found in this published paper and these lecture notes.