I am fairly sure this has been academically discussed in philosophy. Nevertheless, this is something that I have came across so often that I thought it was worth writing about.

Imagine you thought of the possibility of an event, you assign that event $X$ a probability $p$. The most straightforward intepretation is to state that $P(X) = p$.

However, this interpretation is very likely a (less obvious) mistaken.

A simple example

Imagine you secretly admire someone (a crush / secret idol / whatsoever).

One day you are on your way to work / school, and by chance, you bump into that person. You are a very happy human-being. Looking back, you would realize that the odds of this happening was really low. You make a fair estimate for the probability of this event happening – let’s call this $p_x$.

Consider the alternative scenario: you wish to bump into that same person unexpectedly as part of your daily routine. By doing some estimation and research, you come up with a probability of this event happening, let’s call it $p_y$.

Do $p_x$ and $p_y$ refer to the same value?

What?

I would argue that $P(X) = p_x$* and $P(X \space | \space the \space thought \space of \space the \space event) = p_y$. That is, simply by considering the event’s existence, you have subconsicously chose a different probability value to estimate. It is somewhat paradoxical – in both cases, you do not explicitly rule out the existence of event $X$ - but in the latter, you explicitly considered it.

How can we circumvent such biases? This is a very classic example of narrative fallacy. It is incredibly difficult to erradicate this subconscious behaviour. Nevertheless, it still helps to know that it exists, as we can take measures to prevent cathostrophic consequences due to overconfidence / underconfidence in the probability of such events.

I believe expectation scenarios similar to the one presented above is far more common than most of us would care to admit (myself included).

* You can even go one step further argue that $P(X | event \space has \space happened \space before) = p_x$, but the key point here is that the sole thought of an event’s existence alters estimation results.