Chapter XVII · 38 min

Paradoxes of Probability

Where intuition reveals its hidden assumptions

“The theory of probabilities is at bottom nothing but common sense reduced to calculus.”

Probability paradoxes are rarely contradictions in mathematics. They are diagnostic instruments. Each one exposes a hidden assumption: what was sampled, what information was revealed, which population was mixed, what “uniform” meant, or whether an average is being asked to carry more meaning than it can bear. This chapter collects the classics in one place and, more importantly, sends each puzzle back to the idea that resolves it.

Monty Hall: information has a mechanism

Three doors hide one car and two goats. You choose one. Your door has chance $1/3$; the two-door complement has chance $2/3$. Monty knows the location of the car, always opens a goat door, and always offers the switch. He is therefore not removing a door at random: he is transmitting information according to a rule.

If your original choice was wrong — probability $2/3$ — Monty is forced to leave the car behind the other closed door. Hence switching wins with probability $2/3$. The 100-door version makes this almost impossible to unsee.

The birthday paradox: collisions grow by pairs

With 23 people, the probability that at least two share a birthday already exceeds one half (under the usual 365-day, independent, uniform approximation). Compute the complement:

Bertrand's paradox: “uniform” over what?

Choose a random chord of a circle. What is the probability it is longer than a side of the inscribed equilateral triangle? The sentence is incomplete. Choosing endpoints uniformly on the circumference gives $1/3$; choosing the midpoint uniformly along a radius gives $1/2$; choosing the midpoint uniformly over the disk gives $1/4$.

There is no contradiction: these are three probability measures on three procedures. “Random chord” does not identify one of them. Bertrand's puzzle is therefore a philosophical jewel for a book called Measure of Chance: a probability is not merely a set of outcomes; it is a set of outcomes plus a measure.

Simpson's paradox: aggregation can reverse the sign

Suppose A succeeds in 9/10 mild cases and 30/90 severe cases; B succeeds in 80/90 mild cases and 2/10 severe cases. A is better inside each severity group, yet overall A is 39/100 while B is 82/100. B received mostly easy cases; A received mostly hard cases.

The paradox is a warning that weighted averages can reverse comparisons. Conditioning and aggregation do not commute. A lurking variable changes the mixture weights.

The inspection paradox: long intervals are easier to meet

Arrive at an irregular bus stop at a time unrelated to the schedule. The gap you find yourself inside tends to be longer than an ordinary gap: a 20-minute interval occupies twice as much clock-time as a 10-minute interval and is therefore twice as likely to contain your arrival.

This is length-biased sampling. The same skeleton explains why a randomly chosen student reports a larger class than a randomly chosen teacher, and why an ongoing process looks surprisingly old. Sampling objects uniformly and sampling moments uniformly are different experiments.

St. Petersburg: infinite expectation, finite willingness to pay

A fair coin is tossed until the first tail. If the first tail occurs on toss $n$, the game pays $2^n$ units. The probability of that event is $2^{-n}$. Every possible level therefore contributes exactly one unit to the expected payoff:

Yet few people would pay an arbitrarily large entrance fee. Daniel Bernoulli's famous resolution was to distinguish money from utility: an extra thousand matters less to a millionaire than to someone with almost nothing. Modern treatments add other cautions — finite bankrolls, bounded games, risk and tail behaviour — but the central shock remains: expectation is a mathematical average, not automatically a fair human price.

The prosecutor's fallacy: reversing the conditional

Suppose a forensic pattern occurs in only one person in a million. It is tempting to say: “If the defendant matches, there is only a one-in-a-million chance they are innocent.” That swaps $P(\text{match}\mid \text{innocent})$ for $P(\text{innocent}\mid \text{match})$. They are not the same number.

The posterior also depends on how many plausible people could have produced the evidence and on the prior odds. A rare match can be powerful evidence without magically becoming a posterior probability of guilt.

The two-envelope paradox: where did the expectation change its meaning?

Two envelopes contain amounts in a 1:2 ratio. You choose one and see an amount $x$. A seductive argument says the other envelope is equally likely to contain $x/2$ or $2x$, so its expected value is $(x/2+2x)/2=1.25x$; therefore you should always switch. But the same argument applies after switching, producing an impossible perpetual preference.

The hidden error is that after observing $x$, the cases “$x$ is the smaller amount” and “$x$ is the larger amount” are not automatically equiprobable. Their probabilities depend on a prior distribution for the unknown amounts. Without such a model, the conditional expectation is not defined by the story.

One cabinet, one diagnostic habit

• Monty Hall: how was the evidence generated?
• Birthday: what are the true collision opportunities?
• Bertrand: what does “uniform” mean operationally?
• Simpson: what mixture was aggregated?
• Inspection: what does the sampling scheme over-represent?
• St. Petersburg: is expectation the quantity we actually care about?
• Prosecutor: did we reverse a conditional or forget a base rate?
• Two envelopes: did we smuggle in a prior while pretending not to?