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Suppose you are asked to pick any random real number. Then you have a choice to pick any number between -∞ and +∞, i.e, infinite numbers. The probability that you select a particular number n = 1/∞ = 0. Which means that you did not select any number. How is it possible?

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3 Answers 3

Two misconceptions might be at play here: that to pick something at random always means picking something uniformly at random; and that to pick an object at random means that one picks any specific object with positive probability.

Regarding the first point, consider that one cannot pick uniformly at random, say, a positive integer, since the probability $p_k$ to pick $k$ would not depend on $k$ in $\mathbb N$ and the common value $p$ of these probabilities $p_k$ should solve the equation $\sum\limits_{k\in\mathbb N}p=1$, which has no solution.

A way out of this conundrum is to use nonuniform distributions, like a geometric distribution or a Poisson distribution or any other from a plethora. Then $p_k$ does depend on $k$ and one can achieve the condition $\sum\limits_{k\in\mathbb N}p_k=1$.

Regarding the second point, consider the uniform distribution on the interval $[0,1]$. A random number $U$ chosen according to this definition is such that $x\leqslant U\leqslant y$ with probability $y-x$ for every $0\leqslant x\lt y\leqslant1$. In particular, $U\leqslant\frac12$ happens with probability $\frac12$, $\frac13\leqslant U\leqslant\frac12$ happens with probability $\frac12-\frac13=\frac16$, and so on, but $U=x$ happens with probability zero, for every $x$.

This kind of remark, and others, led to the idea that additivity was a desirable feature of a probability, but only when restricted to countable collections.

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The logic you have described only works (still, informally) with countable sets. That is why a measure-theoretic approach was needed to formalize the probability over general sets. Namely, you define the class of "obserable events" (formally, a $\sigma$-algebra of subsets of the original set) and define probability for these events only.

This probability is a measure and it formalizes the notion of being random. Namely, if you ask me to choose a random number, I can choose $0$ or $\sqrt{2}$ with probabilities $0.5$ for each of them, since being "random" is not a formal notion. The formal notion is a probability distribution.

Even for countable sets, how would you define randomness? Namely, it's clear that if we consider a finite set with $n$ elements then the a "natural" interpretation of being "random", I guess, means that any element can be picked up with probability $\frac1n$. But what if there are infinitely many elements? You can still forget about measure theory and define just a probability to pick up each of these elements. But how should one define the probability to pick up $1$ or $2$ or $n$ from $\mathbb N$ in a "random" way?

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probability of selection of any random real number is one...

intuitively, if you have to choose any real number between 1 and 5, you practically don't have an option of not choosing that number. which means you will choose a number between the finite defined range. since there is no other condition on the properties of the number being chosen, the probability of choosing any real number between 1 to 5 is 1.

same soln. can be extended to the infinite case...

a more germane case is putting some property on selection, example- choosing a multiple of 5 on the real number [1,5]. here, the probability is undefined since there are infinite real nos. between 1 and 5. as above, this can be extended to the infinite cae with same results.

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