Britannica Sources used in this answer

How is choosing a random number from zero to one related to tossing a fair coin infinitely many times?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

sample space for a pair of dice — probability theory Sample space for a pair of dice.

Choosing a random number from the interval is mathematically equivalent to the experiment of tossing a fair coin infinitely many times. This relationship is established by representing the sequence of coin toss outcomes as a real number in base 2.12

If we let $X_{k}$ represent the outcome of the $k$ th toss (where $X_{k} = 1$ for heads and $X_{k} = 0$ for tails), the sequence of tosses $(X_{1}, X_{2}, \dots)$ corresponds to the number:
$\sum_{k = 1}^{\infty} \frac{X_{k}}{2^{k}} = \frac{X_{1}}{2^{1}} + \frac{X_{2}}{2^{2}} + \frac{X_{3}}{2^{3}} + \dots$

In this model, any specific sequence of $n$ outcomes has a probability of $1 / 2^{n}$ . This corresponds exactly to the length of the interval $[i_{1} / 2^{1} + \dots + i_{n} / 2^{n}, i_{1} / 2^{1} + \dots + i_{n} / 2^{n} + 1 / 2^{n}]$ in the unit interval. Because both models assign the same probabilities to these intervals, they are considered mathematically consistent. This connection is a fundamental example of how probability theory utilizes measure theory to handle experiments with infinite sample spaces.123