Why do we need sigma-algebras to define probability spaces?

We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) are made up of events to which a probability measure $\mathbb{P}$ can be assigned. Certain properties are fulfilled, including the inclusion of the null set $\varnothing$ and the entire sample space, and an algebra that describes unions and intersections with Venn diagrams. Probability is defined as a function between the $\sigma$ -algebra and the interval $[0,1]$. Altogether, the triple $(\Omega, \mathscr{F}, \mathbb{P})$ forms a probability space. Could someone explain in plain English why the probability edifice would collapse if we didn't have a $\sigma$ -algebra? They are just wedged in the middle with that impossibly calligraphic "F". I trust they are necessary; I see that an event is different from an outcome, but what would go awry without a $\sigma$ -algebras? The question is: In what type of probability problems the definition of a probability space including a $\sigma$ -algebra becomes a necessity? This online document on the Dartmouth University website provides a plain English accessible explanation. The idea is a spinning pointer rotating counterclockwise on a circle of unit perimeter: We begin by constructing a spinner, which consists of a circle of unit circumference and a pointer as shown in [the] Figure. We pick a point on the circle and label it $0$, and then label every other point on the circle with the distance, say $x$, from $0$ to that point, measured counterclockwise. The experiment consists of spinning the pointer and recording the label of the point at the tip of the pointer. We let the random variable $X$ denote the value of this outcome. The sample space is clearly the interval $[0,1)$. We would like to construct a probability model in which each outcome is equally likely to occur. If we proceed as we did [...] for experiments with a finite