The mind likes narratives. Both on an existential level and an everyday level our mind loves to make stories. We'll see a bunch of data points and try to connect them. It's what we do. The problem is as humans it seems our confidence in the validity of a story is a function only of the number of points that fall within the margin of error of that particular story. We seem to lack an intuitive understanding of incorporating the total number of data points into the set, meaning that in cases where there are a large number of data points we tend to be overly confident.
I believe a large generator of false hope is due to a confluence of this overconfidence in the presence of a large amount of information and a lack of understanding of Bayes' theorem. Namely, just because a series of events are not likely does not indicate that something must be causing them, you instead have to compare the probability of the unlikely events with the probability of what you're trying to measure. For example, it's really hard to not overdiagnose some condition that is uncommon. Say some test is 99% accurate. If you have some condition it will tell you that you have it 99% of the time and give you a false negative 1% of the time. Similarly, if you don't have the condition it will give you a false positive 1% of the time. Say also that this condition affects 1% of the population. Say also that everyone in the population would be tested. 00.99% of the total population would be correctly noted as having the condition, 00.01% of the total population would have a false negative, 98.01% of the total population would be correctly noted as not having the condition, and 00.99% of the total population would have a false positive. This means that, of the people who test positive, only 50% have the condition, making the condition significantly overdiagnosed, even with 99% test accuracy.
The problem of overconfidence in narratives is also present on a more personal level in terms of what we believe about our own identities. We have so many memories that the range of narratives we can form - in effect, we can make ourselves whoever we want to be. So much about our personality, our identity, is completely mutable. It's quite weird to think about.