--On Thursday, September 20, 2001 2:02 PM -0400 Matt Cramer <cramer at unix01.voicenet.com> wrote:
> On Wed, 19 Sep 2001 lweiger at umich.edu wrote:
>
> Hmmm, I'm not sure what you mean by the distinction between "individual"
> and "collective" lottery play. Collective play would be a pool, where you
> and 11 mates pick 500 numbers and split the cost of the tickets. That is
> the same as you picking 500 numbers as far as probability is concerned.
> Whether it is worth it to do that depends on your disposable cash and the
> reward and the number of mates in your collective, things that are
> orthogonal to the probability of winning.
I should've used "cumulatively" instead of "collectively." My bad.
> But forget about lotteries for a second, and consider coin tosses. For a
> coin toss, if we want to probability of heads, we divide the total
> favorable outcomes, 1 (there is only 1 way for a coin to come up heads)
> by the total number of outcomes, 2 (either heads or tails). The
> probability of getting a heads is 1/2.
Yep.
> If we do 5 coin tosses, there are now 5 favorable outcomes (a heads on
> any toss), but there are also 10 possible outcomes (a head or tail on
> any toss). The probability of heads is 5/10, or 1/2. Still the same.
>
> Do you follow this with coin tosses? Each coin toss, no matter how many
> times you do it, has a 1/2 probability of coming up heads. It doesn't
> matter if your previous 500 coin toss where heads (See Stoppard's play
> _Rosenkrantz and Guidenstern are Dead_...), the next coin toss still has
> the same probability.
Yep. However, your're more likely to get heads once by flipping the coin multiple times. This has nothing to do with the (non)influece of past events on future events. Your chance of not getting heads at least once in five tries (or, in other words, getting tails five consecutive times) is 1/64.
> The probability works exactly the same way for lotteries, even though the
> fraction is more like 1/10^8. The way which probabilities work doesn't
> change when you are dealing with astronomical odds.
>
> Many folks believe that coin tosses and lotteries have memories; that each
> play somehow "remembers" what it did previously, and that if a certain
> outcome hasn't yet occurred, the toss or lottery will adjust it through
> some unknown means so that the unoccurred outcome becomes more likely.
> But that is silly; lotteries have no memory, nor do coin tosses.
>
> Hence lotteries are taxes on people that are bad at math. :)
I've already rejected the absurd idea of unconnected past events influencing future events out of hand.
> > Even better than playing a million times would be purchasing every
> > combination (discounting the fact that doing so generally results in a
> > net loss of money on the purchaser's part).
>
> As I've shown above playing a million times does not help. But the latter
> method WOULD help, as you are increasing the number of favorable outcomes
> (different numbers you've purchased) and keeping the total possible
> outcomes the same. But that isn't practical, as you indicate.
As someone pointed out to Max, the probability of winning the lottery at least once during the course of a million tries is approximately .63. That's much better than your odds of winning by playing a single time. My expository prose may be sloppy, but the mathematical assertions have been correct.
Here's a real humdinger: if you're on a game show where there's a prize behind one door and x number of booby prizes behind the other doors, you should always switch from your original choice (at least, if you want to heighten your chances of winning) after the booby prizes have been revealed and only two doors remain (the one you chose and the one that probably has the prize behind it).
-- Luke