How does Pr(an event next year) $= 1/100$ imply Pr(at least one of these events occurring in the next 25 years) $> 1/5$?
Please see the embolden phrase below. It appears to equalize, or at least relate, Pr(an event next year) $= 1/100$, with Pr(at least one of these events occurring in the next 25 years) $> 1/5$? But the former is an equality, whilst the latter an inequality?
Turning to other extreme events, we find that there are opportunities for developing long-term contracts that take into account the behavioral biases and heuristics utilized by decision makers. For example, the standard annual bonus system practiced by many organizations could be modified so that bonuses are contingent on multi-year performance. This might induce managers to more systematically consider the potential consequences of their immediate actions over time and to pay more attention to worst-case scenarios rather than hoping that they will not arise by the end of the current year. In the same vein, presenting probabilities of extreme events in the context of a multi-year horizon may lead individuals to pay attention to the resulting outcomes. For example, rather than providing information in terms of a 1-in- 100 chance of an event occurring next year, one could indicate that the chance of at least one of these events occurring in the next twenty-five years is greater than 1 in 5.
Paul Slovic, The Irrational Economist (2010), page 271.
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tldr– It's a persuasive-writing issue, not a math issue.
If an event has a 1% chance per year, then it has a ${ \left( 1 - 0.01 \right) }^{25} \approx 0.7778 ,$ or ~77.78%, chance of not happening in 25 years, and so a ~22.22% chance of happening at least once in those 25 years.
That's more than one-in-five.
The author seems to argue that such a statistic might be more compelling to an irrational reader (given the book's title).
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