Seeing the latest argument in this thread reminds me of the debate earlier on it about the lottery. I noticed no one distiguished between instantaneous probability (single trial) vs probablity over a number of trials. An example would be using a 4-sided dice (yes, I could have used the more common 6-sider but that forces me to use rounded numbers for decimals). One roll has only a 1 in 4 chance of a three showing up, or a 3/4 chance of it not showing up - 75%. TWO rolls gives only a 9/16 or only 56.25% of it not showing up (3/4)^2. At THREE trials the odds for it NOT appearing drops to 42.1875% (3/4)^3, and so on. So the actual odds of winning the lottery by an individual is actually based on the number of times x people play it not just the computed chance of a specific number combination showing. That's why you actually see winners almost every year or so with major wins. The number of people playing actually improves the odds of winning for everyone (increases the number of trials). But what would I know? I just tutor in multivariate calculus and statistics.
I don't specifically recalling a discussion point where this fact is significant. I don't recall (I could be wrong) someone asserting what the odds of
someone winning are, which is based on how many entries are put in and what their distribution are. However, this statement:
The number of people playing actually improves the odds of winning for everyone
is false. I hope you didn't teach that to anyone. The odds of *someone* winning at all is based on the number of *different* powerball combinations that have been played. Since there are 292,201,338 possible combinations, the odds of *someone* winning on a particular draw if N *different* combinations have been entered is N/292201338. If every single combination has been put in at least once, then the odds are 1.0, or a certainty.
What you're talking about is something different. Suppose there are N different entries in the Powerball, and the rules stated that the Powerball operators were to continue to draw powerball combinations *until* someone won. In that case, the odds of the first one generating a winner are N/P (where P = 292... ). The odds that the first draw does not generate a winner would be 1-N/P. The odds that D successive draws would not generate a winner are (1-N/P) ^ D, meaning the odds of a series of draws not generating a winner get lower over time; the odds of drawing a winner increase with more draws.
However, that's not how powerball is played. Hypothetically speaking, if you bought tickets using the computer to randomly pick the numbers, then the odds of you having the winning number
assuming you never draw a duplicate number series ever are 1 - (1-N/P)^T where T is the number of tickets. In effect, given those criteria each random draw is in effect an attempt to hit the winning target randomly, and the odds of never hitting the target follow the same formula. However, the only way to do that is to randomly select your ticket numbers yourself, rejecting duplicates. The lottery computers will not do that for you.
The more people that play, the greater the chances of *someone* winning. But it doesn't increase the chances for
everyone to win. Your individual odds are the same no matter what. However, how many people play does have an effect on your statistical return per ticket, because more players both increases the size of the jackpot if you win and increases the probability you'll have to share that jackpot if you win. But unless you coordinate your efforts with those other people, it doesn't affect your personal odds of actually winning at all.