Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.

Question

My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation?

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Context

Some probability questions have answer $\frac{1}{2}$, and - as you might expect - have an intuitive explanation, i.e. an explanation that requires little calculation, instead utilizing a clever framing of the question that makes the answer obvious.

For example:

"Flip seven unbiased coins; what is the probability of getting at least four heads?"
Some people will calculate $P=\frac{\binom{7}{4}+\binom{7}{5}+\binom{7}{6}+\binom{7}{7}}{2^7}=\frac{35+21+7+1}{128}=\frac{64}{128}=\frac{1}{2}$, but there is an intuitive explanation: getting at least four heads means getting more heads than tails, which, by symmetry, has probability $\frac{1}{2}$.

Another example:

Taking Seats on a Plane and its intuitive explanation.

Another example, this one about geometric probability:

"Break a stick at two random points. The probability that the longest piece is at least twice as long as each of the other pieces is $\frac{1}{2}$. Why?" Here is a (somewhat) intuitive explanation.

On the other hand, I have found that some probability questions have answer $\frac{1}{2}$ but do not seem to have an intuitive explanation. For example:

• The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac{1}{2}$. (Update: Several explanations, of varying degrees of intuitiveness, have been given here.
• Draw tangents at 3 random points on a circle to form a triangle. Show that the probability that a random side is shorter than the diameter is $\frac{1}{2}$. (Update: An intuitive explanation has been given here.)
• Intuition is silent: Find the probability that the smallest circle enclosing $n$ random points on a disk lies completely on the disk, as $n\to\infty$.
• Break a stick at $n$ random points. What is the probability that the three shortest pieces can form a triangle, as $n\to\infty$?

I find these later kinds of questions interesting, because the answer, $\frac{1}{2}$, is the simplest probability of all (unless you count $0$ or $1$, but those are sort of degenerate), but there does not seem to be a simple explanation. It's as if the question is making fun of us: "You can't find an elegant solution!" Or maybe there is no elegant solution, and the answer is $\frac12$ by coincidence.

I would be interested in more of these kinds of questions (they don't have to be related to geometry).

Remarks

I require that the answer is exactly $\frac{1}{2}$, not any other number, to avoid a slippery slope of what numbers are acceptable.

Let's avoid ad hoc questions, for example: "Find the probability that two uniformly random points inside a unit square are within $d$ distance of each other, where $d$ is the smallest positive root of $\frac12x^4-\frac83x^3+\pi x^2-\frac12=0$ ($d\approx0.5120$)." The answer is $\frac12$ because $d$ is the median distance. Obviously anyone asking this question already knows that the answer is $\frac12$.

I am using the "big-list" tag, whose description reads "Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling..." I think my question might provide some insight on probability and intuition.

Answer 1

A triangle's vertices are three uniformly random points on a circle. The side lengths are, in increasing order, $a,b,c$.

We have $P(a^\color{red}{1}+b^\color{red}{1}>c^\color{red}{1})=1$, which is just the triangle inequality.

We also have $P(a^\color{red}{2}+b^\color{red}{2}>c^\color{red}{2})=\frac14$, which has an intuitive explanation.

So what is $P(a^\color{red}{\sqrt2}+b^\color{red}{\sqrt2}>c^\color{red}{\sqrt2})$ ?

The answer is $\frac12$, but an intuitive explanation seems to be lacking.

Answer 2

Regarding the "Intuition is silent"-problem:

The smallest possible circle has the diameter equal to the distance of the two points furthest from each other. In the case that the circle is defined by three points, said diameter will be arbitrary close to said distance, as with infinite points, both cases converge to each other.

With $n\rightarrow\infty$ they both will be arbitrary close to the disk's edge. Looking only at the second point, it will be arbritrary close to the diameter of the disk through the first point. As the distances are arbitrary small, the arc segment becomes a straight line and it forms two right angles with the diameter intersecting it. Now, wether or not the circle will be on the disk depends on the distances of the point to the disk's edge and to the diameter. If the former is larger than the latter, the circle will be on the disk. If not, it won't. Geometrically, this probability corresponds to the point being on one or the other side of the angle bisectors of the right angles. So it is $\frac{1}{2}$.

Answer 3

For almost all real numbers, their binary digits are evenly distributed i.e. the chance of a random digit to be $0$ (or $1$) is $1/2$. More rigourously, the probability of a uniformly randomly chosen digit from the first $n$ binary digits of the number is $0$ (or $1$) approaches $1/2$ as $n$ approaches $\infty$.

A short way to say this is that almost all real numbers are normal in base 2. A more general theorem is that it is true for all bases. I chose this problem because normality, despite intuitive as it may sound, is exceptionally hard to prove for real numbers. We currently don't even know if $\sqrt{2}$ is normal or not.

Answer 4

A triangle's vertices are three uniformly random points on a circle. The side lengths are, in random order, $a,b,c$.

The triangle inequality tells us that $P(a+b<c)=0$.

But what is $P\left(a+b<\left(\frac{a}{b}\right)c\right)$ given that $\frac{a}{b}>1$ ?

The answer is $\frac12$, but an intuitive explanation seems to be lacking.

Answer 5

A triangle's vertices are three uniformly random points on a circle. The side lengths are, in random order, $a,b,c$.

It is known that the probability that the triangle is acute, is $\frac14$. That is, $P(a^2+b^2<c^2)=\frac14$.

What happens if, in the inequality, we replace one of the $c$'s with $(a+b)$ ? That is,

What is $P(a^2+b^2<c(a+b))$?

The answer is $\frac12$, but an intuitive explanation seems to be lacking.

Answer 6

A triangle's vertices are three uniformly random points on a unit circle. The shortest side length is $a$. The diameter of the incircle is $d$.

What is $P(a^2<d)$?

The answer is $\frac12$, but an intuitive explanation seems to be lacking.