having trouble moving from the concept of increasing/decreasing sequence of real numbers to increasing/decreasing sequence of events. I have no problem with this concept regarding real numbers but events is a bit abstract.
My question is could someone provide any real-life examples of increasing/decreasing sequences of events and not in terms of a mathematical definition which I already know.
For example, suppose we have a sequence of coin tosses and we denote event An as the event of getting heads of the nth coin toss. Is the reason why this sequence is an increasing sequence of events is because the indicator function of the event is increasing? What causes a sequence to be classified as increasing/decreasing? Again, not looking for a mathematical definition of why or why not an event sequence is increasing/decreasing but examples and intuition that I can't seem to find in any probability book.
Thanks so much!
Best Answer
Sequences of events don't completely line up conceptually with sequences of real numbers. An event is a set of outcomes. However, the probability of each event in a sequence will create a sequence of real numbers.
An increasing sequence of events is $A_1\subset A_2\subset A_3\subset \cdots$. This does create in increasing sequence of probabilities $P(A_1)\leq P(A_2)\leq P(A_3)\leq \cdots$. For a decreasing sequence of events, just flip the subset symbols and the inequalities.
Regarding coin flips, an example of a decreasing sequence of events could be letting $A_n$ be the event that the first $n$ flips are all heads but that the remaining flips can be anything, so that we have $A_n\supset A_{n+1}$ for all $n$. If $\omega\in A_{n+1}$, then $\omega$ is a sequence of coin flips where the first $n+1$ flips are all heads. Thus $\omega\in A_n$ as well.
An example of a increasing sequence of events could be letting $A_n$ represent the event that after the first $n$ coin flips we only get heads but that the first $n$ flips can be anything. If every flip after the $n^{\text{th}}$ is heads, then every flip after the $(n+1)^{\text{th}}$ is also heads, thus $A_n\subset A_{n+1}$.
One thing to note here is that we are either fixing an increasing number of the coin flips in the first example and thus have a decreasing number of "free" coin flips, and thus a decreasing number of total outcomes included, hence a decreasing sequence of events. In the second example, we are fixing a decreasing number of the coin flips and thus have an increasing number of free flips, hence an increasing sequence of events.
This particular intuitive understanding contained in the above paragraph may not translate to every situation though. However, generally, an event can be thought of as restricting the outcome of a random experiment to a certain set, so an increasing sequence of events means we impose fewer and fewer restrictions/constraints on the experimental outcome and thus are including more and more individual outcomes.