Lotteries and the Limits of Rationality

lottery, rationality

I’ve played the lottery once in my life. I was living in California, and a friend thought it would be fun if we all bought a ticket for the next drawing. And guess what? I didn’t win. This came as no surprise, because I expected not to win. In fact, any rational person who buys a lottery ticket should believe that her ticket is a loser. But oddly, this rational belief leads us into believing a contradiction. Maybe this tells us something about the limits of rationality.

Winner, Winner?

Here’s principle of rationality #1:
If something is 99.99% probable to be true, then you should believe it.

Seems right, doesn’t it? Suppose there are one million tickets in a fair lottery, and I manage to buy all but one. I would have a 99.99% chance of winning. In other words, I should believe that I have the winning ticket. Perfectly rational. The opposite seems equally true. If I have one ticket, I have .0001% chance of winning, and I should believe that I will not win the money. In fact, if I could get every ticket holder to stand in a line, I could walk to each, in turn, and say, “You’re ticket won’t win.” Because for any ticket, the chance of winning is .0001%, and thus I should consider it a loser.

Conjunction Junction

Here’s another principle of rationality #2:
If you believe two or more statements, then you should believe the conjunction of those statements.

lottery, rationality

‘Conjunction’ just means connecting the statements with an ‘and.’ E.g., I believe that “cats are evil,” and I believe that “dogs are good.” So, I should believe that “cats are evil and dogs are good.” Easy peasy. But now we have a problem. Because if I believe, for every lottery ticket, that it will not win, then I should believe the conjunction of all those claims. I.e., I should believe that, “ticket 1 is a loser, and ticket 2 is a loser, and ticket 3 is a loser, . . ., and ticket 1,000,000 is a loser.” But that insanely long run-on-sentence, taken as a whole, is patently false, and I know it is. How? Because I know that one ticket is a winner–that’s how a lottery works!

No Contradictions!

But here’s principle of rationality #3:
If two statements contradict one another, I should not believe them both.

So, rationality demands that I should not believe any contradiction. But it also demands that I should believe both of these contradictory statements: (A) none of the tickets will win & (~A) one ticket will win. C’mon rationality, wth?! Something must have gone wrong here–either with our cognitive processes or with our system of rationality.

What’s the takeaway here? For one, it’s a fun puzzle to play with, if you like puzzles. But you can also resist the paradoxical conclusion and try to find the error in the process. The solutions I’ve seen so far still require me to deny something that seems very rational. Perhaps we’ll find a way to work this out in a fully-satisfying, rational way. But whatever the outcome, I take away a renewed humility about my own rational thought processes and about rationality itself. Our minds, as well as our systems of logic, serve us well, but not perfectly.

As a Christian theist, this neither upsets nor worries me. I know already that even the most brilliant human is still only a human, finite and flawed. The pressure to solve the mysteries and problems of the universe does not finally rest on our shoulders. This doesn’t mean we ever stop wrestling with the problems, though. We will explore and experiment right up to the limits of our powers, straining until we reach our full potential. But if, at the end of the day, there are unsolved mysteries and lingering questions, I’m OK with that. My ultimate trust is not in a rational (and fallible) mind, but in a good God.

4 thoughts on “Lotteries and the Limits of Rationality

  1. Great blog. It reminds me that we must be careful to not base our faith in Christ because someone really smart or famous, says they believe or do not believe in Christ. Instead, it must be because we have personally made the decision.

    1. Thanks, Aaron. I think lots of things can serve to strengthen our faith, but I agree that it ultimately must be something we are personally convinced of.

    2. Hmm. I believe in climate change because that’s what the climate scientists tell me. But I do agree (as an atheist) that believing or disbelieving in God on the basis of somebody else’s authority is bogus. Different stakes involved, I suppose.

      It does seem like the nature of faith does require a personal choice. What were the reasons for your choice?

  2. This might be old news for you Aaron, but this particular puzzle can be resolved with contextualism. Replace principle #1 with “if the reasons subject S has for believing p rules out all relevant alternatives to p being true, then S is justified in asserting p.” Conversational context (allegedly) determines what alternatives are relevant. (Note that I’m using ‘asserting’ instead of ‘believing’ for reasons that will become clear below.)

    In a lot of conversational contexts, you are justified in asserting that the lottery ticket won’t win. “Hey, honey, we could totally buy that cool thing! We could use our lottery winnings to pay our rent!” “Come on honey, you know that ticket’s not going to win.” “Sigh, yeah.”

    Bringing up that for each ticket, you are justified in believing it won’t win, but that the conjunction of those beliefs entails a contradiction, changes the context. In that new context, you’re forced to admit that you don’t *know* that a given ticket won’t win, even though the odds are very much against it winning.

    To bring this back around: the standards of the norm of assertion (knowledge) vary from context to context (assuming contextualism is true). But something I think hasn’t been brought up in the literature yet is that the norm of *belief* remains invariant across contexts. You might not be permitted to assert the contents of your belief, and your belief no longer counts as knowledge, even subjectively. But you are still ‘permitted’ to retain your belief in the new context. “Okay, fine, I don’t know that this ticket is a loser. But I still believe it is.”

    And a lot of religious beliefs at least seem to fit this paradigm…

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