Infinite Outcomes§

Some experiments could conceivably have infinitely many outcomes.

Flipping a Coin

Sps. we had an experiment where we flipped a coin until we get tails, and we record the number of flips required as the outcome, then

$$ \Omega = \mathbb{N}. $$

The outcome is $k$ if the first $k-1$ flips are H and the $k$th flip is T. So, this is one in $2^k$ equally likely outcomes when we flip $k$ times.

$$ \forall k \in \mathbb{N}, P(k) = 2^{-k}. $$

Then, we have

$$ \begin{align*} 1 &= P(\infty) + \sum_{k=1}^{\infty} 2^{-k}\\ &= P(\infty) + 1\\ P(\infty) &= 0. \end{align*} $$

Notice that

$$ \sum_{k=1}^{\infty} 2^{-k} = \sum_{k=0}^{\infty} 2^{-k} - 1 = 1 $$

Geometric Series

In the previous example, we can notice that the infinite summation is actually a geometric series defined by

$$ \sum_{k=0}^{\infty}aq^k = a + aq + aq^2 + \dots = \frac{a}{1-q} $$

where $|q| < 1.$

The probability that we never flip tails is zero, but it is not inconceivable. From the example, we conclude that probability zero does not necessarily equate to logical impossibility.


Sps. $x$ is chosen randomly from the unit interval $[0, 1].$ What is the probability that $x \in [a,b] \subseteq [0, 1]?$

As all locatuions are equally likely, it is trivial to see

$$ P(x \in [a, b]) = \frac{\text{length }[a, b]}{\text{length }[0, 1]} = \frac{b - a}{1 - 0} = b - a. $$

Types of Infinities§

There exists different infinities, generally countable and uncountable infinities.

Countable Infinity

By definition $\mathbb{N}$ is countable.

Any infinite set is countable if each element can be labelled by a natural number, i.e. there exists a bijection $f: A \leftrightarrow \mathbb{N}.$

Discrete State Space

In probability, any space $\Omega$ with finite or countably infinite number of elements is considered to be a discrete state space.

Discrete spaces satisfy additivity:

$$ \forall A \subseteq \Omega, P(A) = \sum_{\omega \in A} P(\omega) $$

Fact: $\mathbb{R}$ is uncountably infinite (i.e. no bijection to $\mathbb{N}$ exists), and so is any interval of $\mathbb{R}.$