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Playing scratch

There’s been a secret Santa, maybe more than one, busy around the Chelsea Green offices the past week. Every morning we arrive at work to discover some little treat at our desks. The first few mornings it was candy, then yesterday a pencil, and today a “Bah Humbucks” scratch ticket from the Vermont Lottery. I won $10!

Sandi asked me if I was going to take the cash or turn my ticket in for more tickets. Being an irredeemable nerd, I decided to calculate the expected return on the game before choosing how to take my winnings.

So at least for this particular scratch game, the expected return on a $2 ticket is an average of $1.50215825. Meaning, if you play it over and over, you will win an average of a buck fifty for every two bucks you spend (aka, 75 cents back for each dollar spent). Of course, those winnings come in lumps, so the thrill can be pretty cool and maybe well worth the price of admission.

Now historically I’ve been more of a Powerball type player than a scratch player. So now I’m wondering what the expected return is on a Powerball ticket. Let’s mosey on over to their website and see… This is a little tricky, because the grand prize is always changing day to day, and it turns out to have a pretty major effect on your expected return. For example, today’s listed grand prize is $53 million. In that case, the expected return on a $1 ticket is 55.9 cents. (I’m including the odds of winning the grand prize as well as the odds of winning the lesser prizes.) If the grand prize gets up to $100 million, your expected return rises to 88.1 cents. The break even point relative to the Bah Humbucks scratch ticket is when the Powerball grand prize is $80,938,548 — that’s the point at which the scratch ticket is (statistically speaking) worth exactly as much as the Powerball ticket. If the Powerball grand prize falls below that amount, your best bet is the scratch ticket; if Powerball goes over that amount, it is your best bet.

But all that is based on the Statistics 101 rule of “an infinite number of plays.” A person can’t play these games an infinite number of times. So the real odds are a little different. And given that the Powerball game sucks in its customers with the promise of an otherwordly huge grand prize (that comes with such impossible odds) yet is very stingy with its lesser prizes, while the scratch has a lower grand prize but is more generous with its lesser prizes… um, this sentence is getting too complicated for me; I’ve gotten lost in my grammar. In other words, Powerball is a sucker’s game. While the overall expected return is roughly equivalent to the scratch, its expected return is heavily tilted towards the grand prize, which, quite simply, you and I are never going to win. The odds of winning the grand prize are 1 in 146,107,962. (The expected return of Powerball excluding the grand prize is a measely 19.7 cents on the dollar; the expected return on the scratch ticket excluding the top prize is still a relatively respectable 73.4 cents on the dollar.) If you bought 10 Powerball tickets every day for 50 years, you’d have bought a total of 182,500 tickets (costing you $182,500). You’re odds of winning the grand prize in all that time would be 1 in 800. That’s not very good.

I guess the moral of the story is that I’m a rube just like everyone else who gets impressed with a big shiny thing like a huge Powerball payoff. It’s time for me to start playing scratch.

Which brings up another question I’ve had for a long time: how does the lottery compare to buying life insurance? Well, I can’t find useful numbers in my quick Google search, and I really do have other work to do, so we’ll have to leave that question for another time. Stay tuned!


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