The Lottery Paradox
A philosophical conundrum where one does not believe any specific lottery ticket will win but does believe that some ticket will win, highlighting the inadequacy of all-or-nothing logic for probabilistic statements and the need for degrees of belief.
THE IDEA
Chapter 19
Popper quote: From Conjectures and Refutations, pages 95–97.
The lottery paradox: This is one example of scholars not understanding the high-impact rare event. There is a well-known philosophical conundrum called the “lottery paradox,” originally posed by the logician Henry Kyburg (see Rescher [2001] and Clark [2002]), which goes as follows: “I do not believe that any ticket will win the lottery, but I do believe that all tickets will win the lottery.” The quotation as supplied says that all tickets will win. The usual lottery paradox instead assumes that some ticket must win while each particular ticket is very unlikely to win; the distinction matters to the argument.
Taleb continues: To me (and a regular person) this statement does not seem to have anything strange in it. Yet for an academic philosopher trained in classical logic, this is a paradox. But it is only so if one tries to squeeze probability statements into commonly used logic that dates from Aristotle and is all or nothing. An all or nothing acceptance and rejection (“I believe” or “I do not believe”) is inadequate with the highly improbable. We need shades of belief, degrees of faith you have in a statement other than 100% or 0%.
One final philosophical consideration. For my friend the options trader and Talmudic scholar Rabbi Tony Glickman: life is convex and to be seen as a series of derivatives. Simply put, when you cut the negative exposure, you limit your vulnerability to unknowledge, Taleb (2005).
When to use it
Evaluating unlikely investment outcomes without treating a small probability as zero.
Making medical decisions based on test results with low base rates.
Interpreting statistical evidence in court cases.
Deciding whether to buy insurance for rare but catastrophic events.
FROM THEORY TO EVERYDAY LIFE
In practice
Fair Lottery with 1 Million Tickets
In a fair lottery with 1 million tickets and a guaranteed winner, someone may reject a win for ticket #123456 because its chance is 1 in a million. They may make the same judgment about every individual ticket while accepting that some ticket wins. The tension arises if they combine all those individually plausible rejections into the belief that no ticket wins. Low probability alone is different from impossibility.
Rare Disease Diagnosis
A low pretest probability can make a rare disease unlikely for a particular patient while the disease still occurs somewhere in a large population. That pair of claims is compatible. A low probability should not be silently converted into a categorical claim that the patient cannot have the disease.
A MOMENT TO REFLECT
Put it to work.
In what areas of your life do you treat low-probability events as impossible, and how might that thinking be flawed?
How can you adopt a mindset of degrees of belief rather than absolute certainty?
Can you think of a situation where 'all-or-nothing' thinking caused you to misinterpret a probabilistic situation?
References
- Paradoxes: Their Roots, Range, and Resolution — Rescher, Nicholas (2001)
- Paradoxes from A to Z — Clark, Michael (2002)
- Conjectures and Refutations — Popper, Karl R.
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)