Probability Spaces

A probability space is the formal container for randomness. It specifies possible outcomes, which outcome sets count as events, and how probability is assigned before random variables, conditional probability, or expectation can be defined.

Definition

A probability space is a triple $(\Omega ,\mathcal{F},P)$:

$$\Omega =\text{sample space},\mathcal{F}=\text{sigma-algebra of events},P:\mathcal{F}\to [0,1].$$

The measure satisfies $P(\Omega )=1$, $P(A)\ge 0$, and countable additivity:

$$P\left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }P(A_i)\text{for disjoint }{A}_i\in \mathcal{F}.$$

For finite sample spaces, $\mathcal{F}$ is often the power set. For continuous spaces, $\mathcal{F}$ matters because not every informal subset is measurable.

Worked computation

For two fair dice, the sample space has $6\times 6=36$ equally likely ordered outcomes. Let $A$ be the event “sum is at least 10” and $B$ be the event “the dice are doubles.” The event $A$ contains six outcomes:

Sum	Outcomes
10	$(4,6),(5,5),(6,4)$
11	$(5,6),(6,5)$
12	$(6,6)$

So $P(A)=6/36=1/6$. The doubles event has six outcomes, so $P(B)=6/36=1/6$. Their intersection contains $(5,5)$ and $(6,6)$, hence $P(A\cap B)=2/36=1/18$. Therefore

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)=\frac{6}{36}+\frac{6}{36}-\frac{2}{36}=\frac{10}{36}\approx \mathrm{0.2778.}$$

The union calculation works because the event algebra contains complements, intersections, and unions; without a fixed $\Omega$ and $\mathcal{F}$, the phrase “probability of either event” would not be well-defined.

Caveats

Most mistakes come from changing $\Omega$ mid-argument. Conditioning on “flagged users” uses a different reference population than all users, and a density on $\mathbb{R}$ assigns probability to intervals rather than exact points.

References

• Probability space
• OpenStax Introductory Statistics 2e, Chapter 3 introduction

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