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.

Intervals

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 $\mathbb{R}.$