Scratch 2 Win vs Lottery Odds

I have a great interest in gambling odds. Not gambling itself, just the theory and mathematics behind it. So I was intrigued by the Sun-Times Scratch 2 Win game that's running for 8 weeks in February and March.

It's a simple "scratch off" type of game. You get a card in the Wednesday paper. There's a scratch-off section for each day, two for Sunday. Each day, you check that day's paper and get the 5 numbers of the day. Scratch those numbers off to reveal an amount under each. If 3 of the 5 have the same amount, you win that amount. There are 4 possible amounts to win: $100, $500, $1,000 and $25,000.

The odds of winning on a weekly basis are listed on this web site:
$25,000: 1 in 420,000 (1 winning card is distributed per week)
$1,000: 1 in 70,000 (6 per week)
$500: 1 in 70,000 (6 per week)
$100: 1 in 13,548 (31 per week)

Therefore, the expected value based on a cost of $0.50 to buy the Wednesday paper to get a card and assuming you could find out the winning numbers for free is:
-0.50*(419,986/420,000) + (25,000 + 6000 + 3000 + 3100)/420,000 = -.4999833+0.0833
= -0.41665

That means for every $0.50 you spend to buy a Scratch 2 Win card (theoretically, I'm sure no one actually does it), you would expect to lose 42 cents. I.e. for every dollar you bet on Scratch 2 Win, you could only expect to get about 16.7 cents back. Those are pretty low odds.

How do those odds stack up against the Illinois State Lottery games, especially the scratch-off games?

I looked on the Illinois Lottery web site and found that they don't list the precise odds for their scratch-off games. However, looking at their annual report for fiscal 2007 (the latest available at the web site), I found some interesting facts:

• Scratch-off games are by far the biggest revenue producer for the lottery.
• These games brought in about $1.040 billion in revenues (51% of total revenues) and they paid out about $707 million in prizes. There are also $38.7 million in prize liabilities for instant games.

With this information, we can make a rough estimate that the expected value of the Illinois Lottery instant games is approximately (707+38.7)/1040 = 0.717. So for every $1 that is played in the Illinois Lottery instant game, you can expect to get about 72 cents back.