On paradoxes in probability

To my taste, popular discussion of the well known paradoxes in probability often misses the point, so let me explain some of my own thoughts. The word paradox is used rather broadly, and I categorize some uses below.

Your intuition is wrong (YIIR)

Textbooks and popular science books describe iconic "Bayes rule" examples: false positives and the prosecutor's fallacy and the famous Tversky-Kahneman taxicab problem. Here YIIR is manifestly true: human minds do not automatically implement Bayes rule in these contexts! A second standard context where YIIR is manifestly true is "coincidences are more likely than you think", inevitably illustrated by the birthday problem. Psychology literature, and notably Kahneman's wonderful popular account Thinking, Fast and Slow, discuss how these and other cognitive biases arise. I tend toward a rather simplistic answer:

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.

Incompletely specified model

The mathematical setup of Probability is designed so you can tell when a mathematical model is completely and consistently specified, in which case any numerical question within the model has some definite answer. But it does not tell you how to go from a verbal description of a question to the mathematical model. This is the case for examples like the Bertrand paradox and the two envelopes problem, discussed from another viewpoint here. The point is that such paradoxes immediately go away when you completely specify the model; how the chord is chosen in the former example, and a prior distribution on the envelope amounts in the latter. Somewhat related are observations claimed to be paradoxes based on some fundamental misunderstanding. For instance Parrondo's paradox does not recognise that the mathematical notion of "fair game" is martingale.

Fantasy stories

I am puzzled by the habit of embedding an asserted paradox into a fantasy story, exemplified by the aptly-named Sleeping Beauty problem, because if you cannot find a more realistic story to make a point, then the point ipso facto can hardly be relevant to the real world. More about fantasies.

Some paradoxes I find interesting

Various other topics in these pages could be considered paradoxes, for instance that you will eventually lose money on a favorable game if you are too greedy. The style of paradox I find most interesting involves questions we can say in words but cannot translate into precise mathematical questions and the linked page gives three examples.

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:
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.
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.

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.

Left: in a simulated 16-player single-elimination sports tournament, the green curve shows the probability that the top-ranked team is the winner, as the lower curves do for selected other ranks. The x-axis parameter indicates spread of skill, best interpreted by the blue curve showing the probability the top-ranked team beats the second-ranked team. Right: in a simulated 300-player 100-question prediction tournament, the distribution of rank of winner, centered around the 60th ranked.
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, the winner of a prediction tournament is most likely to be one of the good-but-not-best cohort . This is illustrated in the right figure above, taken from and explained more in this paper.


This figure shows the ``effective number" of names for boy/girl babies born in the U.S. in different years. Here "effective number" relates to the birthday problem. In particular for girls born in 2000 this number is around the number of days in a year. So for a group of young ladies who were born in 2000, the chance some two have the same birthday is roughly the same as the chance some two have the same given name.