It's easy to assume that those who buy lottery tickets have no understanding of mathematics, or at least no understanding of probability. These idiots are just letting the allure of a giant jackpot overpower their understanding of the odds. Why would anyone buy a ticket if they knew their chances of winning were infinitesimal?
Well, as it turns out, the gut instinct that motivates lottery players-- "Sure, I have almost no chance of winning, but if I do win, I could be a millionaire!"-- has a little mathematical support. One of the most useful formulas for everyday decision-making is the Expected Value Formula, pictured below:
In English, the idea is that the expected value of any random variable X (for example, the amount of money you might win from the lottery) can be calculated by multiplying each possible value of X (you win $5, you win $10, you win $100...) by that possibility's respective probability (there's a 10% chance of winning $5, a 3% chance of winning $10, a 1% chance of winning $100...) and add all those products together. Even if the chance of winning the jackpot is small, a big enough jackpot can outweigh the cost of participation.
Obviously, it's a little complicated to explain in the abstract, so let's use a simple example.
I'm at the carnival, and I come across a game of chance. The toothless man in the booth says that if I give him a dollar, I get to choose one letter from the word "GAMES," and if he pulls my letter out of his hat, I'll win ten dollars. Each letter has an equal probability of being drawn. Should I play?
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| Pictured: Trustworthiness |
There's definitely a greater chance that I'll lose than win-- 80% chance of losing versus a 20% chance of winning. But ten dollars is, like, two more rides on the Tilt-a-Whirl, or two and a half funnel cakes. I'll risk my dollar for a shot at some more fried dough.
Four out of five times, I'm going to lose my dollar. One out of five times, I'll win ten dollars, leaving me with a net gain of nine dollars. So, we calculate:
(0.8)(-$1) + (0.2)($9) = -$0.80 + $1.80 = $1.00
Hey! My expected value is positive, meaning I should totally give this guy my money! Well, in strict mathematical terms, it means is that if I continue playing this game indefinitely, stuck in some terrifying carnival-based Groundhog Day loop, my losses and winnings will balance out to a net gain of $1. In practical terms, a positive outcome from the expected value formula means that the cost to play is low enough and the prize is high enough that this miniature lottery is definitely worth playing.
Of course, things don't work out so well for Mr. Carny. Four out of five times, he gets to keep my dollar, but one out of five times he has to give me back my dollar plus nine more out of his pocket. His expected value from this venture is:
(0.8)($1) + (0.2)(-$9) = $0.80 + -$1.80 = -$1.00
Not a great way to run a business. This guy needs his meth money, so he decides to change the rules of the game. Now, it costs me $3 to play, and I win $13 for picking the right letter, leaving me with a net gain of $10. I still have the same chance of winning, and even though the game costs more to play, I could win a larger sum of money than I could before. But before I fork over my cash, let's do the numbers on this new version of the game:
(0.8)(-$3) + (0.2)($10) = -$2.40 + $2.00 = -$0.40
It's no longer beneficial for me to play. Someone paying $3 and picking a letter is more likely to lose money than gain anything in the long run, but our crafty gamemaster is counting on most folks' inability to grasp the subtleties of expected values. He can paint bigger prize numbers on his signs while still bringing in more money than he pays out.
Let's leave behind our oversimplified example, and move on to the real money.
The North Carolina Powerball isn't the only lottery game available in the state, but it's the one with the highest current jackpot ($130,700,000 cash value as of Feb. 5th) so it's the example I'll pick. It also has an intriguing "PowerPlay" option, which doubles most available prizes for just one extra dollar up front. I want to see if it's worth it, from an expected value standpoint, to PowerPlay.
The NC Education Lottery homepage has links to detailed explanations of each of their games, as well as summaries of the odds of winning any particular prize in any particular game, and the number of real-life people who've actually won each prize to date. I'm using information from the Power Ball "How To Play" page, and right off the bat, I have a slight issue with how they're displaying the "odds" of winning.
A lot of people toss around "odds" and "probability" thinking they mean the same thing, but they're two different ways of expressing the likelihood of an event. Probability (the values we need for the Expected Value Formula) is a fraction or percentage, while odds are generally represented as a ratio.
Those numbers in the "odds" column, as you can see, don't look like either. They're actually the total possible outcomes for each combination of numbered balls. It gets a little complicated to explain how the lottery folks generate those numbers (white balls numbered 1 through 59, red balls numbered 1 through 35, order doesn't matter, disjoint sets...) and we won't get into that right now. Maybe later!
To make use of those numbers, we can read the "odds" column as "one divided by this number equals the probability of you winning such-and-such particular amount." And when you add them all together, they total to the "overall odds of winning" figure at the bottom (1 out of 31.85), so we can assume that the overall odds of not winning anything are 30.85 out of 31.85 (or, if you like whole-number numerators and denominators, 617/637). These are the values we'll use for our calculations.
Each Powerball ticket costs $2, so we'll subtract 2 from each possible prize to reflect the fact that we're already out $2 for the cost of the ticket. Drumroll, please!
Well, shucks. Turns out that playing the lottery has a negative expected value! Who would have guessed? Even though the jackpot is REALLY BIG, the enormous probability of losing outweighs the value of the possible winnings. Also, that $2 cost significantly eats into your winnings for the smaller, more likely prizes, further dampening your expected value outcome.
But hey, what about that enticing PowerPlay option? By paying just one extra dollar, I could double or even quadruple my winnings! It's not tough to substitute the PowerPlay option into our formula: all probabilities stay the same, while only the winnings change. Go big or go home! Mama needs a new pair of shoes!
Looks like mama's going barefoot for a little while longer. The PowerPlay option actually decreases your expected value. You'll wind up losing more money in the long term by choosing it, though the Lottery Board ends up making a little more money, which explains why they offer it.
The moral of the story is this: while it is mathematically possible, and sometimes even probable, to make money playing a lottery-style game, you usually won't. You'll just end up funding North Carolina public education, and who wants that?
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