All Above 31

Amazingly, all the balls in yesterday’s $250 million Mega Millions drawing were above 31. As I mentioned before, this had happened only four times before in the fifteen-year history of the Powerball and Mega Millions lotteries.

The winning numbers were 37, 40, 48, 53, 56, with Megaball 44.

I did have a ticket for this drawing, even though I only buy a couple lottery tickets a year. All the numbers on my ticket were above 31, as usual, but I matched only two balls: 37 and 40.

Nobody matched all six balls, so I wasn’t the only loser. The prize pool rolls over to Friday’s drawing, with an estimated jackpot of $325 million ($191 million cash value).

Lottery again

The multi-state lotteries are in the news again. On Saturday an Indiana Powerball ticket won the $314 million jackpot ($142 million cash value). There is a $250 million ($146 cash value) Mega Millions drawing tonight.

But the reason I’m writing is because I read in a New York Times blog that “studies and statisticians have varied opinions over the precise long-shot odds of actually winning a huge lottery prize.”

What??

Of course, the odds of winning the jackpot are precisely known. Each Mega Millions ticket has a 1 in 175,711,536 chance. Each Powerball ticket has a slightly better 1 in 146,107,962 chance. There are no dissenting opinions.

The quote might make sense if we’re talking about continually buying tickets and cumulative odds with respect to some ‘huge’ threshold amount, or if we’re considering something like the possibility that a winning ticket will have to split the jackpot.

The author compares lottery odds to death-by-lightning odds. Perhaps he intended to say that statisticians have varied opinions over the precise long-shot odds of being killed by lightning. Now that I can believe.

[edit: The story has been amended and now reads “Studies and statisticians give varied calculations for the precise odds of actually winning or splitting a huge lottery prize.” emphasis added]

Long odds

Subway restaurants have a Scrabble promotion going on. It is similar to McDonald’s Monopoly promotion in that some game pieces are instant winners (usually for food items) and some must be collected into sets to be redeemed for more substantial prizes. To win a grand prize in the Scrabble promotion, one must form the word subway by collecting six individual letters: S, U, B, W, A, and Y.

What caught my eye is the odds. According to the official rules the odds of claiming a grand prize are about 2.2 billion to one. Up to 10 grand prizes will be awarded, they say, but at those odds I don’t see how. The contest is limited to legal U.S. residents, of which there are only about 300 million. So if each legal resident plays the game 7 times we would expect only one winner, right?

Also, the grand prize is only $100,000. The odds of winning MegaMillions or Powerball lottery jackpot are a dozen times better (at 176 or 146 million to one) yet the prize values greatly exceed $100,000. (A lottery ticket is cheaper than a large Subway beverage, too, though it admittedly lacks quenching value.)

The odds of winning the Monopoly grand prize are 20 times longer at 41 billion to one, but at least the prize is $5 million (paid over 19 years, so closer to $3 million cash value). $100,000 is Monopoly’s fourth-largest prize, at 700 million to one odds. On the other hand, the odds of winning food prizes are 1 in 7 for Monopoly, versus 1 in 2 for Scrabble.

Perhaps McDonald’s and Subway calculate their odds differently. McDonald’s official rules say, “the odds of winning Collect & Win prizes are based upon obtaining the complete Winning Combination,” which is vague. Subway doesn’t say anything at all, which is even more vague.

If I recall correctly—and possibly I don’t—in previous years the Monopoly official rules listed the odds of obtaining the single rare piece in each set. I find that more informative, though I suppose it could be assailed as underestimating the true odds. In the Scrabble promotion, it’s unclear if there is a single rare piece in each set.

Lottery Denouement

The jackpot prize pool for Tuesday’s Mega Millions lottery drawing turned out to be $233 million, corresponding to an advertised jackpot of $390 million. The unclaimed jackpot prize pool for the March 2nd drawing was approximately $164.3 million.

The jackpot prize pool is 31.8% of sales so, at a doller per a ticket, 216 million tickets must have been sold for the March 6th drawing to account for the $68.7 million difference. That’s a lot of tickets. Presuming that the numbers on those tickets were chosen uniformly, we’d expect 1.23 of those 216 million tickets to exactly match your ticket. Even if your crazy time-travelling uncle told you the winning numbers in advance, you would expect to be splitting the jackpot two or three ways.

$400 million Lottery Jackpot

The estimated jackpot for tonight’s Mega Millions state lottery (California, Georgia, Illinois, Maryland, Massachusetts, Michigan, New Jersey, New York, Ohio, Texas, Virginia, and Washington) drawing is up to almost $400 million.

The lottery has deservedly been called a stupidity tax, but how does this record jackpot change things? The odds of any specific ticket winning are 175.7 million to one. That gives a naive expected value of about $2.25 for each $1 ticket. Not bad, eh? Why not take out a short-term $175.7 million loan, purchase 175.7 million different tickets, and make a nice quick profit of $220 million or so? It’s easy money.