Okay, your standard lottery game has a fixed number of possibilites. So, buying every single possible combination, while difficult and time consuming its quite possible. Now lets make this into a nice formula:

Profit = Winnings - Number of Tickets/Combinations x 1 Dollar

So if there are say 10 million possible combinations, and the payout is say 50 million, you'd be crazy not to do this. Granted it does have some problems, if more than one person gets the right number you have to split the money, so thats always a consideration. This particular scheme though, probably won't make you very popular with your Lottery Commision. I remember seeing this put to work on a TV show once, I think it was bunch of Australians (In New York) who did it, they got a couple of investors to help buy all the tickets, and used computers to generate the ticket slips and bought them at a convience store for some reason. Unfortunately the store stopped accepting them after a point, so they had a large gap of numbers. They won anyways though.

Actually, you're almost guaranteed to lose money that way, because lotteries are done for profit, which means that the money they pay out is less than the price of all the tickets combined. Usually, it's about half as much.

conform makes a good point in his wu - Most lottery jackpots are progressive, meaning that when nobody gets the winning combination, the money is added to next week's jackpot. This means there can be situations when there's more moeny in the pot than it would cost to buy a ticket for all possible combinations. However, a big jackpot also makes the lottery more attractive, so you're more likely to have to share the money.

Brazil, it's not that simple. As lottery jackpots are progressive, you can wait until the odds are in your favor (if they ever actually are -- read on). While the folks who run the lottery are certainly getting their share of the money, that doesn't mean that it's impossible to have the odds in your favor. Let's consider a simple example:

Suppose you've got a big line of people. A million people, even. You join the end of the line. Each person, in turn, is allowed, if they wish, to pay one dollar to play the game. The house takes half the dollar and pockets it. The idiot (oops, I mean player) flips a coin. If it comes up heads, the player take the pot. If it comes up tails, they lost, the fifty cents they put in is added to the pot, and the next player comes up.

The theory of mathematical expectation tells us that this is a pretty crappy game for the players. The first guy up will, on average, lose 75 cents -- half the time he wins 50 cents, the other half the time he wins nothing (on his dollar investment). Same for anybody who plays the game with nothing in the pot. If you play, and there is 50 cents in the pot already, you expect to lose 50 cents -- half the time you win a dollar, half the time you win nothing. If the pot has $1.50 in it, you're playing a break even game. The fourth guy in line will be in this position roughly one in eight theoretical-runs-though-the-line.

Now, periodically (roughly once every 21,000,000 times, the line will get all the way to you wihtout anyone winning. There will be $500,000 in the pot. For a dollar investment, you'll have a 50/50 shot at winning $500,000.50. You would have to be a collosal fool not to take this bet. If you made this bet over and over again, you would expect in the long run to end up with over $250,000 for every time you played the game!

The lottery is a little different, of course, because lots of people get to play at the same time, and if multiple people win, you split it. Suppose that in our example above that you and a friend both get to play when it comes to you. You both pay a dollar. There are now four equally likely outcomes -- neither of you win, you win, your friend wins, you both win. You win nothing half the time, all of it one-fourth of the time, and half of it one-fouth of the time, for an expectation of $187,500.38. If three people play, your expectation drops to $145,833.77. If enough people are playing, your expectation can drop back into the negative -- meaning you shouldn't play at all (assuming you have a decent idea of how many other people are going to play).

I don't have the time or the figures to work out the circumstances under which lottery play would be favorable, but suffice it to say that they don't come very often. It doesn't mean they can't, however. Zero sum and negative sum games often have moments when the odds tilt towards a particular strategy. The key to taking advantage of this is to have enough information to be aware of it, and to be flexible enough to step in and out when circumstances change between favorable and unfavorable states. That is why blackjack, normally a negative sum game, can in some circumstances be beat for a small profit, and why casinos take the precautions they do -- playing through small portions of multiple decks, requiring regular betting patterns, etc. It's also the essence of casino poker (especially Texas Hold'em) -- the casino is siphoning money out of the game, and you are trying to find the odds in your favor often enough that you can win money faster than the casino takes it away. Lotteries, however, suck enough money, have a high enough cost of assured victory and sell enough tickets when the jackpot gets big that the likelihood of one or several other persons willing becomes quite high. And that, and not the simple fact that lotteries are negative sum, is why covering every ticket combination is probably not a good way to make money.

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