Before we get into it, I must preface this post by saying that I am a person of average intelligence and not a statistician or a math wizard. A few years ago, I began to take an interest in studying the lottery. This post provides a high level overview of the different games offered, their prizes, and the probability of winning. One area that I wanted to explore, which I don’t think has been explored before, is the probability of winning a large secondary prize, rather than the grand prize or jackpot.
To be clear, this post is not about:
- a strategy for picking winning numbers
- winning number frequency or analysis
- any kind of system or method for “beating” the lottery
Multi-State vs. State Run Lotteries
There are two types of lotteries in the United States. One type is multi-state games, which includes the two most widely-played jackpot games, Powerball® and Mega Millions®. Powerball began in 1988 as Lotto America and took on its current name in 1992. Mega Millions began in 1996 as The Big Game, changing to its current name in 2002.
Both games require the player to match 5 numbers plus a bonus number to win the grand prize. The minimum grand prize is $20 million for the Powerball and $40 million for Mega Millions. An average win is between $95 million to $140 million for the Powerball. The largest lottery win in the U.S. was a Powerball prize worth $1.586 billion, which occurred on January 13, 2016.
The other type is state-run games, which vary from state to state. There are 45 U.S. states that have some version of a lottery. The five states with no state-run lottery are Alabama, Alaska, Hawaii, Nevada, and Utah. In many states, a portion of the proceeds from these games go to benefit public education. The prizes and odds of these games vary from state to state. For this post, I will be focusing on the Arizona Lottery, since that is where I live.
The Arizona Lottery has two divisions or groups of games, which are: Draw Games (examples: The Pick, Pick 3, Fantasy 5, and Triple Twist) and Instant Win Games (aka scratchers or scratch-offs).
What are the Odds?
Opponents of the lottery often quote American author Ambrose Bierce, who in his 1906 book called the lottery “A tax on people who are bad at math.” Proponents of the lottery claim that buying a ticket gives people hope. But what are the real odds of winning the lottery?
The odds of winning a jackpot prize in a multi-state game such as Powerball or Mega Millions are ridiculously, unbelievably, astronomically high. As of 2020, the odds of winning the Powerball Grand Prize are 1 in 292,201,338. The Mega Millions is even less likely, with odds of winning the Grand Prize at 1 in 302,575,350. It is often reported that the odds of being struck by lighting (1 in 12,000 – according to the National Weather Service) are far better than the odds of winning the lottery.
These odds are so fantastic that personally, I do not believe that multi-state games are worth playing at all. And yet, lottery ticket sales continue to trend upward nationally. Lottery sales in the U.S. totaled more $91 billion dollars in 2019, according to the North American Association of State and Provincial Lotteries, or NASPL.
State-run draw games have MUCH better odds of winning, at least in Arizona. However, the prizes are significantly smaller than the multi-state lotteries. Below are the prizes and odds for three Arizona Lottery games: Fantasy 5, Pick 3, and The Pick.
The odds of winning the grand prize in The Pick are 1 in 7,059,052, and 1 in 749,398 for Fantasy 5. These are still high odds, but not as high as the 1 in roughly 300 million odds for the large, multi-state games.
Let’s analyze the prize structure of these games. I was not able to find an official system or classification for the amount of lottery prizes. So, I came up with this one on my own:
|Grand Prize or “Jackpot”||$1,000,001 and up|
|Mid-Tier Prize||$10,001 to $1,000,000|
|Small Prize||$100 to $10,000|
|Consolation Prize||$1 to $99|
After studying the prizes for the two multi-state games and three state run games, I noticed something. The prize structures follow a similar format: a single grand prize, and many winners who get $100 or less. There are very few mid-tier prizes in all of these games.
In Powerball, there is a $1 million prize and a $50,000 prize, then the next tier drops to just $100. Mega Millions has a similar structure, with a grand prize followed by a $1 million prize, a $10,000 prize, and then prizes of $500 down to $2. It’s the same for the state-run games. The second largest prizes are: $2,000 in The Pick, $500 in Fantasy 5, and $250 in Pick 3.
When we think about lotteries, the goal is to win the grand prize or jackpot – a huge sum of money, usually many millions of dollars. I believe that lottery players are enticed to play by these enormous jackpot prizes, which would afford them the dream of quitting their job, paying off their debts, and living out the rest of their life on a topical beach with a drink in their hand, complete with tiny umbrella. A jackpot worth hundreds of millions of dollars is such an incomprehensibly large sum of money to the average person that it may as well be infinite.
I am not a financial advisor, but I believe that the average middle class or working class American’s life would be significantly improved by winning a mid-tier prize – say for example, as little as $50,000 dollars. $300 million is simply too large of a number to wrap your head around.
A one-time win of $50,000 would not allow most people to quit their job and retire in lavish comfort for the rest of their life. But what it would do is dramatically improve the financial situation of the average American person or family.
$50,000 dollars, carefully managed, would allow a person or household to pay off their consumer debt, maybe put a down payment on a house, pay off a vehicle, or start a college fund. It would be a windfall that would completely change their financial position in life.
The typical multi-state game has a prize structure that awards a jackpot of say, $300 million, to a single winner, and prizes of $2 to $500 to a group of a few hundred or few thousand winners.
My thought is that the lottery could be much more impactful with a different prize structure. Imagine that the payout is still $300 million, but instead of awarding that to a single person, the grand prize is capped at a percentage or dollar amount. The grand prize winner still gets a huge award – say $30 million or so (which is more than enough to last a lifetime) – and the remaining $270 million is distributed as Mid-Tier prizes of $100,000 to 2,700 winners. Sure it would be nice to win a mega-jackpot, but for myself and for a lot of households, winning a mid-tier prize of $10,000 to $1 million would be life changing.
Asking the Right Questions
There is no shortage of scientific papers, studies, articles and research on the subject of lotteries. I noticed that essentially every study is focused on the odds of winning the jackpot prize. None of the papers I have come across account for the fact that lottery games has a tiered prize system, where many smaller prizes are awarded as well. After spending some time thinking about the lottery, a number of questions began to emerge:
- Which games have the largest non-jackpot prize? (a mid-tier prize worth $10k to $1m)
- What are the odds of winning a substantial, non-jackpot prize?
- Can an advantage be identified by studying publicly available game data?
These are questions which I believe have not been explored by anyone else. I was not able to find any sources online of people interested in calculating the best odds of winning the largest runner-up lottery prize.
Searching for The Best Runner-Up Prize
To summarize, the draw games from both the multi-state games and the state-run games have a prize structure that has essentially no mid-tier prizes. A lot of people are going to win $5, which is essentially the cost of a ticket and is enough to keep them playing, while a grand prize win is statistically next to impossible. For this reason, I decided to explore the odds of instant win games, also called scratch-off tickets.
Scratch to Win?
The first instant win games were introduced in the U.S. in the 1970s. Today they are widely available, including in Arizona.
I began my research by visiting the Arizona Lottery’s website. They have a list of all of the current scratchers games, as well as the odds of winning, rules, and remaining prizes for each game. That last part is important – did you catch that? The lottery’s website shows which prizes have been won for each scratch off game. This information seems very valuable to know as a player.
What I learned is that the Arizona Lottery usually has between 45 and 60 different instant win games running concurrently. The games are designed to last anywhere from 6 to 24 months before the prizes are exhausted. When one game ends, it is often restarted with a slightly different name or graphics, though the premise remains the same. Sometimes the changes are seasonal, such as a Match-3 game called “Red Hot 7s” or “Crazy 8s” taking on a candy cane or snowman theme during the holidays.
In Arizona, scratch tickets are available in a broad range of prices including $1, $2, $3, $5, $10, $20, $25, and $30 tickets. I believe that the Texas Lottery offers the most expensive instant win game at $50.
In Arizona, I observed that the maximum prize is related to the cost of the ticket. After reviewing the AZ Lottery website, the following prize tier becomes obvious.
|Ticket Cost||Top Prize|
|$20, $25, and $30||$250,000 to $2.5 million|
Where it gets interesting is in the upper range of scratch off tickets. The $20, $25 and $30 tickets have grand prizes of $250,000 to $2.5 million. THIS is the sweet spot for me – the type of prize that would be significant enough to be life changing, but with better odds than the 1 in 300 million chance of winning a multi-state draw game.
So what are the odds of winning on these high-end scratch tickets? Well, they’re actually pretty good in comparison. Take a look at the table below.
|Game||Top Prize||Odds 1 in||Prizes Available||Overall Odds 1 in|
|$20 Million Blowout #1221||$50,000||74,977.12||24||3.39|
|Arizona Gold #1268||$500,000||279,886.7||9||3.08|
|$20 Million Cash Explosion #1190||$2.5 million||1,858,166||8||3.09|
If I had $20 to spend on lottery tickets, I would rather have a 1 in 75,000 chance of winning $50,000 than a 4 in 292 million chance of winning the Powerball. Die-hard Powerball and Mega Millions players will often rationalize their behavior with statements like “You can’t win if you don’t play.” Well to that I would say: statistically, you will not win even if you play every week for your entire lifetime. The odds of a 1 in 292 million event occurring are so absurdly high that it may as well be never.
The Overall odds of winning any kind of cash prize in Powerball are 1 in 24.87 and 1 in 24 for Mega Millions, as shown in the screen shots above. The Arizona State games have overall odds of 1 in 9.62 (Fantasy 5) and 1 in 39 (The Pick). Looking at the table above, we see that the higher-priced instant win tickets have overall odds of approximately 1 in 3 of winning something.
Putting It To The Test
Author’s note: I have been wanting to write a version of this article since early 2016. Some of the events in this article, like the part where I actually buy tickets based on the research I have done, took place in 2015.
So here is the part where I put everything together everything I have learned. I reasoned that the best chance of success at the lottery would be:
- Buy a scratch off ticket in the $10 to $30 range. These tickets have the best odds of winning overall, and significantly better odds of winning the grand prize than a multi-state draw game.
- Use the Arizona Lottery website to identify instant win (scratch off) games that have recently started and still have the grand prize remaining.
- Buy a few tickets and see what happens.
Well, I did just that. I found a game called Crossword Party, which had launched on 9/11/2015.
The game had a total of 20 grand prizes, worth $50,000 each. The odds of winning the grand prize were approximately 1 in 250,000 and the overall odds were 1 in 3.67. I figured that buying four tickets, I should win something.
Scratching the tickets is just a formality, as all you need to do is scan the barcode on the back of the ticket at the lottery kiosk to see if it is a winner. But I wanted the full experience, so I scratched the tickets off. Two of them were not winners, one was worth $10, and the other was worth $20. So it would seem that the odds held true. I actually came out ahead by $10 after my little experiment.
The number one rule is that the lottery is designed as a revenue generator for the state – do not forget that. But if you are going to play, play smart. Don’t blow your money on draw games. Find a scratch-off game with a good size prize, the best odds of winning, and one where all the top prizes have not yet been claimed. Oh, and never play with money that you cannot afford to lose!
Disclaimer: The information in this article is presented for entertainment purposes only, and nothing in here should be used as financial advice.