• Relativist
    509
    I just learned that I won the lottery last night. Although on the surface, one would think that my chances of winning were the same as everyone else, it's not true. I am a 65 year old male of Czech ancestry. The odds are massively against a 65 year old male of Czech ancestry winning because we only comprise .03% of the US population - so my chances of winning were .03%* the average American's chances.

    Obviously, I still had the exact same chance as every American at winning, so my reasoning is problematic. What is wrong with this reasoning? Is it a formal fallacy?
  • fdrake
    1.8k
    The probability of a Czech person winning the lottery in the US is P(winning & being Czech), which is less likely because there are less Czechs than people in the total population of the US (and by assumption less in the lottery ticket purchasing group). But this doesn't mean you're less likely to win given that you've purchased a ticket, because P(winning given being Czech) is exactly the same as P(winning given not being Czech) - winning is independent of nationality.

    You can visualise the first probability statement as randomly drawing people who purchase tickets from the total population - winners who are Czech are less likely.

    You can visualise the second probability statement as stating your chances of winning given that you've purchased a ticket - which are the same as a non-Czech holding a ticket.
  • Terrapin Station
    6.9k
    There might be something that fits better, but it's basically a variation on the gambler's fallacy--using irrelevant factors to determine the likelihood of something.
  • Relativist
    509
    I think you nailed the error. I can express it in terms of conditional probability:
    where P(a|b) = the probability of a, given that b is true

    C=being Czech
    W=winning

    P(W|C)=P(W|~C)=P(W)
  • fdrake
    1.8k


    Yep. And the lessened probability of being a Czech and a winner from the general population of ticket buyers would be P(W & C)=P(W)P(C) and P(C)<1.
  • SophistiCat
    697
    One can read your problem statement as "What is the probability of a 65 year old male of Czech ancestry winning the lottery?" Given that there is at least one 65 year old male of Czech ancestry in the pool of participants (you), and that the chances of each participant are the same (by assumption), the probability in question is at least as high as the probability of any single participant winning the lottery. If there happen to be two 65 year old males of Czech ancestry playing the lottery, the probability of a 65 year old males of Czech ancestry is twice as high as the probability of any single participant winning.

    BTW, congratulations! How much did you win? :)
  • Relativist
    509

    I didn't actually win, and obviously it's because I'm a 65 year old, male Czech.
  • Mww
    295


    Dunno about a logical fallacy, because as soon as you won, it’s all moot anyway, the rationale for not winning becomes irrelevant.
  • NKBJ
    501


    It's further complicated by the fact that the % Czech population in the US is somewhat irrelevant to the likelihood of a Czech winning the lottery. It doesn't tell us what percentage of lottery ticket buyers are Czech. It's possible that it's a similar stat, but it's also possible that Czechs buy or don't buy lottery tickets at greater rates than other parts of the population.

    Also, it's a fallacy of division: attributing to a part what is true of the whole.

    For the entire Czech population the chances may be X to win, and for all the winners there may be x percent chance of being Czech, but you can't attribute those statistics to any individual participant. Like, if I won, my chances of being Czech are nil, because I'm not Czech. The chances of me winning the lottery are also nil (even if I were Czech), because I don't play.
  • Tim3003
    43
    The odds are massively against a 65 year old male of Czech ancestry winning because we only comprise .03% of the US population - so my chances of winning were .03%* the average American's chances.Relativist

    The fact of your nationality (and age) is irrelevent as regards your chance of winning a lottery, so this statement is wrong. The chance that the winner would be a Czech (any Czech) may be .03% of the chance that it would be an American (any American), but your chance as an individual is the same as any other individual's.
  • Relativist
    509
    The fact of your nationality (and age) is irrelevent as regards your chance of winning a lottery, so this statement is wrong. The chance that the winner would be a Czech (any Czech) may be .03% of the chance that it would be an American (any American), but your chance as an individual is the same as any other individual's.Tim3003
    Tim- I know that. I was just looking for a way to succinctly show that it's wrong.

    For the record, this pertains to a discussion I'm having about the so-called fine-tuning argument.
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