Why the Lottery Is a Bad Bet

"Stop for Cash," says the poster on Metrorail. It's not reminding you to use your local ATM; it's coaxing you to buy a ticket in the D.C. Lottery. MDB Communications, the lottery's ad agency, won a Certificate of Excellence for this poster at Washington's 1995 Addy Awards.

Their wording probably was a lot more effective in getting people to part with their money than telling the unvarnished truth: "Stop and lose some cash. Here's your chance to a make a voluntary contribution to the D.C. government."

Most people understand that lotteries give out less money than they take in. That's why the District, Maryland and Virginia run them. What many people don't understand is just how much more the lottery brings in than it gives out. The odds and the payoffs are always set up so that in the long run the lotteries make lots of money, and people who play them lose lots of money. That's the whole point, but somehow it never comes across in the get-rich-quick ads.

You can get a better grasp of this by studying the simple arithmetic of probability by which the lottery works.

When you're analyzing probabilities, the long run is what counts. Think about it this way: If you flip a coin once, your chance of predicting the right outcome is no better than your chance of predicting the wrong outcome. If you flip a coin 200 times, however, you can make a better prediction: it's going to come out fairly close to 100 heads and 100 tails. If you flip it 2,000 times, it's going to come out even closer to 1,000 heads and 1,000 tails.

You can't predict any single flip of the coin, but you certainly can predict that when you have many, many flips, 50 percent will be heads and 50 percent will be tails. In the long run, predicting coin tosses is a sure thing.

If you want to bet intelligently on the outcome of a game of chance, there is a way to figure your odds of winning in the long run. It's called the "mathematical expectation," and once you calculate it, you know where you stand.

If a game is "fair," the mathematical expectation is zero. Your chance of winning is as good as your chance of losing. The two possible outcomes cancel out. For lotteries, the mathematical expectation is negative; you always lose money in the long run -- because that's the way the game is designed.

To find the mathematical expectation, multiply the potential payoff by the probability of winning, then subtract from it the amount you bet. You can get a sense of this in a simple game of heads and tails.

Here's the game: You pay $1 and flip a coin. If it comes up heads, you get $1.50 back. If it comes up tails, you get nothing back.

Your mathematical expectation is the payoff ($1.50) multiplied by the probability of winning (your chance of guessing right is one out of two, which is the same as 0.5) minus the amount of the bet ($1).

Here's the arithmetic:

Step 1: $1.50 x 0.5 = $0.75

Step 2: $0.75 -- $1.00 = -- $0.25

In words, your mathematical expectation is negative 25 cents. It's a losing game. You are bound to lose an average of $0.25 every time you play. And the more you play, the more you lose. Play ten times, and you lose, on average, $2.50. You might lose more and you might lose less, depending on how many heads come up in 10 plays. You'll most likely get the average of five heads, or four or six. You still lose money.

The odds and the payoffs in a lottery are also set up for you to lose in the long run.

For example, in fiscal year 1995 the Maryland State Lottery had total sales of more than $1 billion. The total amount they paid out to ticket holders in their various lotteries was about $400 million. (This information is displayed on the lottery's Web page, at www.msla.state.md.us/msla/ about.htm.)

Where did the rest of the money go? It went to the Maryland treasury, after administrative expenses were subtracted.

The net result is that on average, bettors lost about 60 cents every time they wagered a dollar. They would have lost less money with the heads and tails game described above. The D.C. and Virginia lotteries may have different odds, but you can bet they, too, are designed to make money for the governments.

Some people who buy lottery tickets claim that they do it for entertainment value. That's fine, as long as they view it the same way as putting quarters down the slot in a video arcade. But if they play the lottery because they think they're going to win money in the long run, they're just kidding themselves.

In the long run you lose money. That's what the odds say. That's why states run lotteries. It's a form of voluntary taxation. The more you bet, the more you lose.

Robert L. Hershey, a consulting engineer in the District, is the author of the book, "How to Think with Numbers."