** Most YIIR examples occur because you confuse an uncommon or intangible situation
with a more common or tangible one.**

Human minds presumably evolved to deal with the latter.
For instance,
one common explanation of YIIR in the birthday problem is that you confuse it with the
situation where you consider a match with *your* birthday, which requires about 250
people for a 50% chance of a match.
Pondering "does someone else have a May 5 birthday?" is more tangible than
"are there some two people with the same birthday?".
A less common observation is to compare with the situation where everyone in a group introduces themself by
saying their first name.
If two people have the same first name, this does not seem so surprising.
Memory of grade school suggests it is quite common for two children in a class
of 30 to have the same first name, and mathematics confirms this (see Extras below).
So in a sense the "paradox" is that we don't recognize the birthday setting as being essentially the same
as the "same name" situation we understand from experience.

Other YIIR examples such as the Monty Hall problem and Nontransitive dice deal with rarely encountered situations but are useful in learning the math.

A paradox that deserves to be more widely known is
**A bet may be favorable to both parties.**
This involves a trick, of course, and goes as follows.

Suppose I (resident in the U.S.) and a friend (resident in the U.K.) agree that there is a 50-50 chance that, one year from now, the pound/dollar exchange rate will be on either side of 1.28 dollars per pound. (The figure didn't matter, just that we agree on the figure). We could now make a bet:The reader will quickly see the trick: we are measuring in different units which vary in a way related to the random event in question.If 1 pound is at that time worth more than 1.28 dollars then he will pay me 1 pound; if not then I will pay him 1.28 dollars.So what will happen? From his viewpoint he either loses 1 pound or wins an amount that by definition is worth more than 1 pound; from my viewpoint I either lose 1.60 dollars or win an amount that by definition is worth more than 1.60 dollars; and since we agreed it was a 50-50 chance, to each of us this is a favorable bet.

A final and (I believe) recently noticed paradox involves our running theme of
prediction tournaments.
In any natural model of a sports setting, for a league season or a tournament with one eventual winner,
the *relatively* most likely winner will be the best team.
(How likely this is in *absolute* terms depends on the variability of skills amongst teams).
This is illustrated in the left figure below, taken from and
explained more in this paper.

Consider instead a natural model of a prediction tournament. Regarding one-on-one comparisons between contestant, this is like a sport: the more skillful contestant is likely to perform better. But paradoxically, under somewhat plausible assumptions,