The Lottery Fallacy

Taxonomy: Logical Fallacy > Formal Fallacy > Probabilistic Fallacy > The Lottery Fallacy1

Thought Experiment:

A lottery was organized by a charity for a single prize of a million dollars. Unlike state lotteries in the United States, this lottery had a million numbered tickets that were printed and sold for over a dollar apiece. Fortunately for the charity, all of the million tickets were sold. Lucky Larry purchased one of those tickets.

To select the winning ticket, all one million ticket stubs were placed in a large rotating drum, and then a single winning ticket stub was selected at random from the drum after it had rotated many times. The number on the ticket stub matched the number on Larry's lottery ticket. Larry had won a million dollars!

But wait a minute. "Not so fast!" said a disgruntled player, Loser Louie. Louie contacted the news media and charged that Larry must have cheated. After all, Louie argued, the odds against Larry winning were a million-to-one. He, Louie, had bought a thousand tickets and so was a thousand times more likely to win than Larry.

Somehow the selection of the winning ticket must have been rigged in Larry's favor, Louie claimed. He demanded an investigation of Larry and the lottery. "Larry must be connected to someone inside the charity who had access to the rotating drum", according to Louie. What else could explain Larry's improbable good fortune?

What's wrong with Louie's argument?


Event E is unlikely. (Where E is a simple event2 in a sample space in which every simple event is equally unlikely.)
E occurred.
Therefore, E probably didn't occur by chance; something must have caused E to happen.


It was extremely unlikely that the universe would be hospitable to us human beings, or even to life in general. For instance, if some of the basic constants of physics had been even slightly different, we wouldn't exist. Yet, obviously we do exist. Therefore, the constants of the universe must have been "fine-tuned" to produce us, or at least to produce life from which we could evolve. Those precise constants must have been selected to make a universe that is habitable by human beings. What could possibly be responsible for fine-tuning the universe to create a home for humanity? Clearly, only an enormously powerful and intelligent creator of that universe, that is, what we call "God".3


In mathematics, a sample space is a set of simple events together with the probabilities of those events occurring. For example, consider the familiar situation of flipping a fair coin: the sample space consists of two simple events4―heads or tails―each of which has the probability of one-half. Similarly, the sample space for the roll of a fair die is 1, 2, 3, 4, 5 and 6, each with a probability of one-sixth. In both sample spaces, one and only one of the events can happen and each such event is equally probable. Additionally, the probabilities of the simple events in the sample space sum to 1, meaning that when the coin is flipped one of heads or tails must occur―another way to say this is that the complex event that either heads or tails comes up has the probability of one. Similarly, when a fair die is rolled, one of the numbers from one through six must come up, so that the probability of the complex event {1,2,3,4,5,6} is one.

A lottery of the type described in the Thought Experiment, above, is a sample space, with the events being the selection of each of a million ticket stubs. The probability that any one of the individual stubs is selected is one in a million, but the probability that some one of those stubs is selected is one. It is highly unlikely that any given ticket will be chosen, but it is certain that one will be chosen. Thus, it is certain that a one-in-a-million event will occur.

This seeming paradox lies at the heart of the Lottery Fallacy. It is not genuinely paradoxical because there are two different events:

  1. The event of a given ticket stub being chosen is a simple event in the sample space of the lottery and it has a probability of 1/1,000,000.
  2. The event of some ticket being chosen is the complex event of ticket stub 1 or ticket stub 2 or…ticket stub 1,000,000 being chosen, and the probability of this event is 1.



  1. As far as I've been able to determine, this name comes from the following source: Stephen Law, The Philosophy Gym: 25 Short Adventures in Thinking (2003), pp. 71-72. The name is based on the following example:
    Suppose you buy one of a thousand lottery tickets. You win. That your ticket should be the winning ticket is highly unlikely, of course. But that doesn't give you any reason to believe that someone rigged the lottery in your favour. After all, one of the tickets had to win, and whichever ticket won would have been no less unlikely to win. So there's no reason to believe that your win must be explained by someone or something intervening on your behalf―there's no reason to suppose that you have been the beneficiary of anything other than spectacular good fortune.

    Just before this passage, Law writes that "[p]roponents of the [fine-tuning] argument are often accused of committing the lottery fallacy." I'm not sure whether Law means that the same substantive accusation is often made―as I have done in weblog entries―or whether the accusation is actually made under the name of "the lottery fallacy"―something that I did not do. I wonder about this because I've found no evidence of the term "lottery fallacy" outside of the above passage, except in cases that appear to trace back to it. See: The Lottery Fallacy, 7/9/2011.

  2. A simple event is one that is not complex, that is, it is not constructed from other events; for example, when rolling a die the simple events are rolling a 1, 2, 3, 4, 5, or 6. Complex events are sets of simple ones; to return to the die rolling example, rolling an even number and rolling a number less than four are both complex events.
  3. This is a concise version of what's known as "the Fine-Tuning Argument"; see: New Book: The Fallacy of Fine-Tuning, 6/19/2011 & Fine-Tuning the Fine-Tuning Argument, 6/27/2011.
  4. We'll ignore the possibility of the coin landing on edge; in the unlikely event it should do so, we'll flip the coin again until it lands flat.
  5. The above two paragraphs are based on the following entry: The Arguments that Failed, 1/20/2009.
  6. See, for example, the cosmologist Martin Rees' book, Just Six Numbers: The Deep Forces That Shape the Universe (2000).
  7. See the weblog entries linked in notes 3 & 5, above.

Posted: 4/11/2023, Revised: 5/4/23