Lecture 1. B) Probability Function
Probability Function
A probability function is a function [math]P:\mathcal{B}\rightarrow\left[0,1\right][/math], where: .
- [math]P\left(S\right)=1.[/math]
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- [math]P\left(\bigcup_{i=1}^{\infty}B_{i}\right)=\sum_{i=1}^{\infty}P\left(B_{i}\right)[/math], whenever [math]{B_{1},B_{2},...}[/math] are pairwise disjoint.
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- Notice that we haven't defined the domain [math]\mathcal{B}[/math] yet.
. The properties of the probability function are intuitive. The probability of observing any event in set [math]S[/math] should equal 1, so we interpret [math]P\left(S\right)[/math] as the probability of any event in [math]S[/math] taking place. As for the second property: it states that the probability of a union of sets is the sum of the probabilities of each set, when the sets do not intersect. Clearly, we are interpreting [math]P\left(B\right)[/math] as the probability that any of the events in [math]B[/math] occurs.
Remark
By now, you’ve probably noticed that, rather than talking about probability, we’re talking about sets and set theory. This is not too complicated. It’s simply the approach that mathematicians have taken to formalize probability, from ‘first principles.’ Essentially, we would like the probability function to map any set into a probability, i.e., a number between zero and one.
Let us consider the domain of the probability function now.
An intuitive solution is to make [math]\mathcal{B}=2^{S}[/math], i.e., [math]\mathcal{B}[/math] is the power set of [math]S[/math]:
[math]2^{S}=\left\{ \emptyset,\left\{ Heads\right\} ,\left\{ Tails\right\} ,\left\{ Heads,Tails\right\} \right\}.[/math]
The power set is a ‘set of sets.’ The power set [math]2^{X}[/math] contains all the subsets of [math]X[/math]. Clearly, [math]\left\{ Heads\right\}[/math] is a subset of [math]S[/math]. So is [math]\left\{ Heads,Tails\right\}[/math], because a set is a subset of itself. Finally, the power set always contains the empty set, because the empty set is a subset of all sets.
Let’s now check whether [math]2^{S}[/math] seems like a good candidate for the domain of the probability function, [math]\mathcal{B}[/math]. With [math]\mathcal{B}=2^{S}[/math], we obtain the following probability function for coin tossing:
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[math]P\left(B\right)=\begin{cases} 1, & B=\left\{ Heads,Tails\right\} \\ \frac{1}{2} & B=\left\{ Heads\right\} \\ \frac{1}{2} & B=\left\{ Tails\right\} \\ 0 & B=\emptyset \end{cases}[/math]
We’re making progress: we can interpret [math]B=\left\{ Heads,Tails\right\}[/math] as the event of obtaining “heads or tails”, and we can also interpret [math]B=\emptyset[/math] as observing an 'impossible' event, such as the coin settling down vertically, or observing it land heads and tails. When we consider sets with countable many elements, the power set is indeed the default choice for the domain of the probability function.
