To buy a lottery ticket, or not

Larry, Moe, and Curly are shooting the breeze around the coffee bar one morning when Larry says, "Hey! Today's the 13th. That's my lucky number. I'm going to buy a lottery ticket." Moe, who never buys lottery tickets because he'd rather use the money for haircuts, says, "I understand the jackpot is an estimated $40 million. Lots of people will be buying chances for that." "Yea," says Curly. "That's why I never buy a ticket when the jackpot's a kerbillion dollars. I figure I have a better chance of winning when the take is smaller because fewer people will be buying tickets." At this moment, Moe is certain that Curly's logic is flawed, but can't quite grasp why. While this puzzled trio gulps coffee, let's ask an SwRI Whizard to clear up the confusion.

"Believe it or not, Curly's assumption about the probability of winning the lottery is very common. Even television newscasters have been known to state that one's odds of winning the lottery decrease when more tickets are sold. This is simply not true. Let's try to explain this probability concept with an easy scenario.

"Suppose you have five coins in your pocket: a penny, nickel, dime, quarter, and half-dollar. Your five-year-old daughter asks you to give her two coins. You reach into your pocket and pull out the nickel and the dime. She is somewhat excited, but has recently started learning the face value of money in her kindergarten class and would like a larger amount. So, you return the two coins to your pocket and draw again. This time you give her the dime and the half-dollar. Excitedly, she runs off to tell her brother about her good fortune.

"What are the odds of pulling out the nickel and the dime versus the odds of pulling out the dime and the half-dollar? To answer this question we need to compute the number of "combinations" of two coins we could pull out of our pocket. Remember that the order in which I pull them out of my pocket is not important. That is, getting a penny first and a quarter second is the same "combination" as getting a quarter first and a penny second.

"So how many ways can we select two coins from a total of five? Let's count them: (1) a penny and a nickel, (2) a penny and a dime, (3) a penny and a quarter, (4) a penny and a half-dollar, (5) a nickel and a dime, (6) a nickel and a quarter, (7) a nickel and a half-dollar, (8) a dime and a quarter, (9) a dime and a half-dollar, and (10) a quarter and a half-dollar. There are ten ways to select two coins out of our group of five, so the odds of getting a nickel and a dime are the same as the odds of getting a dime and a half-dollar. Both combinations have a 1 out of 10 chance of being selected.

"Now, how does this relate to a state lottery? There are 50 numbers from which you are asked to select six. How many combinations of six numbers can be drawn from a total of 50? Well, I won't list them all here, but you would go through the same process outlined in the earlier example and count that there are 15,890,700 combinations of six numbers that can be drawn from 50. Therefore, the odds of selecting one 'winning' combination of six numbers is 1 in 15,890,700 (check the back of any lottery play slip for the listed odds). Notice that this probability does not depend on the number of ticket combinations purchased. So, the odds of picking the winning combination of six numbers is the same for a measly $4 million pot or a whopping $50 million pot.

"One more point. While Curly's logic was flawed in his computation of his chances of winning the lottery, his assumption about the influence of lots of people buying tickets has some merit. Remember that the odds of selecting any one combination of six numbers from 50 are the same. We have already agreed that those odds do not change. But what does change when the jackpot gets larger and more people are buying tickets? Well, what changes is the odds that more than one ticket will be sold with the same combination you picked. That is, with more people buying tickets, the odds that someone else selects the same six numbers you selected have increased (assuming that all combinations of six numbers have the same chance of being chosen by the ticket buyers). Buying your ticket when the jackpot gets large simply increases the chance that you will split your winning ticket with another winner (or winners).

"Now, if you feel really lucky, go ahead and pay a dollar and hope that you beat the odds. It's only a 1 in 15,890,700 chance!"

Thanks to this month's Whizard, Janet Buckingham, a principal analyst in the statistical analysis section of the Automotive Products and Emissions Research Division. Buckingham is an experienced statistician, mathematician, and computer scientist who provides her expertise to research programs Institute wide.

The Lighter Side SwRI Home

March 25, 2013