On probability assessments for the longer term future

in what real world contexts is it both practical and useful to attempt to estimate numerical probabilities?

we can ask, in various contexts,

how far ahead is it ever reasonable to attempt any probability assessment?

The various Global Catastrophic Risks mostly familiar to SF enthusiasts are fun to discuss in class, but I have nothing worth writing here, except
(a) the threat of super-intelligent AIs or the Technological singularity strikes me as completely implausible. For a recent discussion see this April 2018 Barry Purcell blog post. For an amusing analysis of the players in this debate see this August 2017 Daniel Estrada blog post.
(b) The most important context is surely the extent and consequences of future climate change. Readers who wish to engage this topic seriously should start with the Fifth Assessment Report (published 2013-14) available via the IPCC site. I will not discuss any specifics of this context. But how might one try to compare this context with other contexts in which people attempt long term forecasting? This page presents some rather fragmentary thoughts.

1. One familiar example concerns world population projections, which currently assert with 80% confidence that the 2068 population will be between 10.0 and 10.5 billion. This topic is conceptually simple -- we know the proximate factors are life expectancy and birth rates -- but then we need to guess how these might change over the next 50 years. One can start with extrapolation and historical correlations between economic growth and fertility, but the interactions between population growth/decline and other possible changes (technology, climate change) are difficult to imagine. Regarding economics, this 2018 paper uses both statistical extrapolation and expert judgments to conclude

Our primary results suggest a median 2010 - 2100 global growth rate in per-capita GDP of 2.1% per year, with an SD of 1.1%, indicating substantially higher uncertainty than is implied in existing forecasts. The larger range of growth rates implies a greater likelihood of extreme climate change outcomes than is currently assumed.

2. Various short-term (up to 3 years, say) probabilities can be gleaned from financial futures markets or insurance premiums or prediction markets. Explicit probabilities for more than a few years ahead are hard to find; I am only familiar with two ongoing programs giving 10-year assessments. Here is a graphic from the 2007 Global Risks Survey showing assessed probabilities of major economic-impact risks of the coming 10 years which contains numerical probabilities; curiously this annual report later switched to qualitative descriptions of likelihood. And an ongoing series A Quick & Dirty Guide to War by James F. Dunnigan and Austin Bay (apparently last published in 2008) does give explicit probabilities for specific potential conflicts over a 10-year window. For instance, on the topic of the Cold War in Europe, their 1985 probability assessment for 1985-1995 was
65% status quo
25% internal revolts in Eastern Europe lead to decrease in Soviet control
5% military attack by Soviet Union on West Germany
5% Soviet Union falls apart for internal reasons
and their phrase "the empire crumbles" for the latter was rather accurate.

3. In contrast, the general style of most attempts at envisaging the future is illustrated by Global Trends 2030: Alternative Worlds (published 2012). This lists 14 items, from ongoing trends to more dramatic possibilities (6 are copied below) without attempting the assess their likelihood.

Individual Empowerment. Individual empowerment will accelerate owing to poverty reduction, growth of the global middle class, greater educational attainment, widespread use of new communications and manufacturing technologies, and health-care advances.
Diffusion of Power. There will not be any hegemonic power. Power will shift to networks and coalitions in a multipolar world.
Crisis-Prone Global Economy. Will global volatility and imbalances among players with different economic interests result in collapse? Or will greater multipolarity lead to increased resiliency in the global economic order?
Impact of New Technologies. Will technological breakthroughs be developed in time to boost economic productivity and solve the problems caused by a growing world population, rapid urbanization, and climate change?
Stalled Engines. In the most plausible worst-case scenario, the risks of interstate conflict increase. The US draws inward and globalization stalls.
Fusion. In the most plausible best-case outcome, China and the US collaborate on a range of issues, leading to broader global cooperation.

4. Another context is provided by the 100 year bond (dollar denominated) issued by Argentina in 2017, and paying 7.9% interest on initial price. For a short term bond or loan, the appropriate interest rate is the risk-free rate plus a premium for estimated probability of default. For the 100 year bond there is some more complicated calculation based on the time-discounted estimated probability of default. This is like the "rationalist" view of stock prices, that they should reflect discounted estimated future profits; though it seems that more investors are willing to follow the rationalist view for bonds than for stocks.

5. A well known Bill Gates quote is

We always overestimate the change that will occur in the next two years and underestimate the change that will occur in the next ten.