The Illinois St. Patrick’s Day Lottery Raffle is about to be drawn and it offers “the best odds of winning a million dollars ever offered by the Illinois Lottery.” That last bit is important – “ever offered by the Illinois lottery,” because it is not at all someone’s best chance at winning $1,000,000. Let’s start with a breakdown of the lottery’s raffle:

There are 500,000 tickets sold at $20 each and the prizes are four (4) $1 million winners, five (5) $100K winners, and five hundred (500) $1K winners for a total of $5 million in prizes for $10 million taken in (which doesn’t include the tax the winners will have to pay back to the state). The four winners out of half a million entries comes out to a 1 in 125,000 chance of winning $1 million. When my coworker brought this up, I declared that betting red or black in roulette however many times in a row it would take to reach $1 million from $20 probably has better odds than this lottery raffle.

Here’s how that betting would work:

First of all, let’s assume there are no maximum bets. To begin, your cash amount = $20 and the odds of getting to next amount by betting red/black = 18/38 = 47.37%

In the 2nd turn, Cash amount = $40, overall odds of getting to next amount= (0.4737)^2 = 22.44%

This keeps going on for fifteen turns, at which point:

Cash amount = $655,360, and the odds you got this far are 1 in 73,734 or 0.00136%.

Also, you will probably get a friendly or not-so-friendly visit from the pit boss and the casino manager for getting your bets right and doubling your money 15 times in a row.

Instead of betting it all on black or all on red, we only bet $500,000 on one color and the other $155,360 on the other color.

There is an 18/38 chance you just won $1 million, an 18/20 chance you still have $310,720, and a 2/38 chance you have nothing. This cycles through a few times where if you lose the last bet, but get the smaller bet correct you keep playing and the odds add in, but in total, the odds for the 16th bet are 68.5% (compared to 47.37% if you only bet one color) to get to $1 million. In total, your odds are (0.685 * 0.0000136) = 0.000929% or 1 in 107,641. So I was right. You have a 16% better chance of winning $1 million by playing roulette and picking colors.

The odds are better if you pick a single number in roulette. You have a 1 in 38 chance of winning and they pay out 35 to 1. If you win 3 times in a row, you would have $857,500. Then you can make thirty bets of $28,575 on individual numbers (with $250 left over to bet on a 31^{st} number). If one of those 30 full bets comes up, you just won $1,000,125. The odds playing this way are 0.00001841% or 1 in 54,308, which is 230% more likely to win $1 million than the lottery raffle.

You might think the best way to go in a casino is by playing blackjack, because it has the “best odds of winning.” However, in order to let a bet ride the way we are doing, you would not be able to double down or split your hand in blackjack and the odds would dramatically decrease.

Now I don’t have anything against playing the lottery. The entertainment appeals to our lizard brains’ greed. If you get the first number right, your brain gets flooded with endorphins and the prospect of winning that much money is worth the near zero likelihood that it will happen. I do, however, have a problem when people try to tell me that the lottery is some sort of smart gamble. In this instance, I proved the lottery is not your best chance of winning $1 million, and I don’t believe that there is ever a point where a regular lottery jackpot is large enough that it is a smart gamble – where the expected value of a $1 ticket is greater than $1.

I can remember reading an article about a specific lottery where at the end of the year, they gave out the built up or unclaimed winnings in a special drawing and this drawing did have a positive expected value for tickets, but I can’t find that article.