Wouldn't it be great if you could win a million dollars? Suppose you had $200 that you could expend to try to win $1,000,000.
What would your best course of action be, expected value be damned? It's not like you're going to spend that $200 thousands
of times, so you don't really care about expected value. You only want the best chance of making the big score, the
$1,000,000 prize. What should you do?
First, I should explain what prompted this article. I read about a house raffle, where people can purchase raffle tickets for
$150 each, and the big prize is a house worth over $1,000,000 (or a similar amount of cash). There are other prizes as well,
but like I said we're not interested in expected value for this excerise, so we'll ignore that. 32,000 raffle tickets will be
sold, so your chances of winning are 1 in 32,000. Sounds better than the lottery, right? Maybe not...
The house raffle certainly sounded better than the lottery to me. But I wanted to find out by how much. So I visited the
California lottery page and what did I see? Coincidentally, there was a note on the web site saying
"Coming Soon: Your best odds at a million bucks." Amazing (what are the odds of that?). There are slightly different
odds depending on how many
people enter, but I'll simplify. Basically, this "Million Raffle" lets you buy 50 raffle numbers for $200 total.
There are 5 million raffle numbers, 10 of which will win a $1,000,000 prize. Since one number has a 1 in 500,000 chance
of winning, 50 numbers gives you a 1 in 10,000 chance of winning. That's much better than the house raffle! (again,
not taking into account expected value)
So then I thought, well, 1 in 10,000 is pretty good. What's the best possible chance, assuming the person offering you
the prize isn't losing money? Well, that's easy. $1,000,000 divided by $200 is 5,000. So the best chance
can't be better than 1 in 5,000.
So 1 in 10,000 isn't really close to 1 in 5,000. Then I thought -- and you'll laugh at first -- roulette wheel! Now,
everyone knows playing games in a casino is a sucker's bet, right? The roulette wheel is probably horrible, right?
Not so fast. Let's say you put that $200 on a single number. You have a 1 in 38 chance of hitting it, and they'll
pay you 36 times your bet, in this case $7200. Now, assuming you can find a casino that allows you to place huge bets,
let's take that $7200 and bet it on a single number again. We'll have a 1 in 38 chance of taking that $7200 and
making $259,200. Great. Now all we have to do is double that twice. We'll bet on black twice, each time having a 18/38
chance of winning. If we win both, we'll have $1,036,800. Our chances of winning are (1/38) * (1/38) * (18/38) * (18/38) =
about 1 in 6436. Surprisingly, much better than the lottery. As I mentioned, you'll have to find a casino willing to
let you bet half a million dollars on one spin. The Plaza Hotel in Las Vegas did allow Ashley Revell to place a $135K bet
on one roll in 2004. He won, doubling his money.
What about craps? You've probably heard that craps and blackjack give the best odds in the casino. I won't discuss
blackjack because it involves a little skill as well, so what about craps? There are lots of different kinds of bets you
can make. Let's analyze a few.
We can bet that the next roll will be 2 (two dice are rolled, and they both come up "1").
If we bet $1 and win, we'll get paid $31. There are 36 possible outcomes of a two-dice roll, with only one of those being 2.
So your odds of winning are 1 in 36. Taking our $200, we'd have $6,200 after one roll, and $192,200 after two.
We can then bet "Any 7", which means that we are hoping the next number will be 7. We'll get back 5 times
our original bet if we win. Our chances of winning are 6 in 36, or 1 in 6. So if we win we'll have $961,000
and our odds will be (1/36) * (1/36) * (1/6) = 1 in 7776. Not very good, and we didn't even make $1,000,000.
Of course, not all craps bets are equal. Other bets at the craps table give you better odds, but are more
complicated to analyze or execute. I'm not an expert craps player by any means (in fact I've never played),
but it looks like your best bet for these purposes would be to find a casino which lets you bet up to 100
times your bet for pass odds. Basically, you make a pass / come bet, then if you win that you're allowed to make
a bigger bet with no house advantage! Obviously, casinos limit the amount you can bet when you do this. Most
won't let you bet more than 5 times your original bet, so if you can find one that allows you to do 100
times, that increases your chances of making the big score.
Note: please keep in mind that this is purely an intellectual exercise. All of the above examples have negative
expected value, so if you continue to spend your $200 over and over again, you'll lose.