Probability vs. Possibility: Lottery Analysis in Quarantine

The probability of you winning is low, but the probability of someone winning is high. The paradox of the lottery.

It’s a scary time right now. Many people are losing their jobs, unemployment is past 20%, there is a global pandemic, and there doesn’t appear to be a clear path back to normalcy.

But the lottery is still being played in most states, with modifications. I know a few different people who play the lottery, and I was surprised to hear that they were still playing the game, despite it all. But as I thought about it more, the more sense it made (or so I thought).

People dream about winning the lottery. “If you had a million dollars, what would you do first?” is a question we’ve all heard before. And now, with the economic outlook looking increasingly bleak, I was curious to know if people still searching (or increasingly searching) for that chance to win it all?

Disclaimer: This is not written in support of gambling, and I do not support gambling. None of this is investment advice, and it is simply analysis about the lottery. If you feel like you might have a gambling problem, please call The National Problem Gambling Helpline at 1-800-522-4700.

Pre-Coronavirus Lottery

Pre-COVID, more than half of Americans played the lottery, according to a 2017 survey conducted by Gallup. 2018 lottery sales topped over \$77.7B, a marked increase from 2017’s \$72.5, according to La Fleur magazine.

Bankrate released the results of a survey describing the ‘vice spending’ habits of the everyday American, with the Silent Generation spending an inordinate amount of money on the lottery, alcohol, tobacco, and gambling.

They found that households in the lowest income bracket (<\$30k / year) spend 13% of their income on lottery tickets. People in higher income brackets spend less of a percentage of their income on lottery tickets, but they still spend a large amount on the game.

The average adult spends \$325 per year on the lottery. According to Bankrate.com, millennials spend an average of \$976 a year on tickets. In the eyes of some, lotteries are considered to be a “voluntary tax” as the proceeds from the purchase go towards the environment or infrastructure or education.

The below pie chart details where the money tends to go. According to a CNN study, about ~63% of the money is paid out in prizes. The other 37% is dispersed across education (~50% of the 37%) state general funds, social programs… and a whopping 34% goes to administrative costs and commissions to lottery retailers.

So Who Actually Plays the Lottery?

According to the Bankrate survey, people in lower income brackets put a larger percentage of their income towards lottery purchases. The lottery has also long been argued to take advantage of poorer groups, acting as a type of regressive tax (if following the notion of a voluntary tax).

Critics of state-sponsored lotteries argue that they disproportionately prey on the hopes of disadvantaged groups such as the poor and less educated, who can least afford to be spending on non-essentials. According to some lottery critics, lottery winnings can have an outsized appeal to economically vulnerable Americans who have less disposable income.

Source: Gallup

However, this isn’t always true.

In fact, the group most likely to buy tickets were in the \$36,000 – \$89,999 income bracket, with 56% of those individuals purchasing tickets. Those with college degrees or some college were also more likely to purchase tickets as compared to their high-school or less educated counterparts.

However, it is important to note that there are two types of lottery players:

1. Those that buy 1 ticket a week
2. Those that buy 10 tickets a week

Gallup did not study the frequency of tickets purchased, just the the purchase of a product of any type. The studies on the population that plays the lottery have a variety of results. There are several different research methodologies, and those yield different research results.

It can appear from the data that higher-income individuals are playing more, but as was shown earlier with the Bankrate study, a large part of the analysis has to do with the amount spent, in addition to the frequency.

Participating in the lottery is the most common form of gambling among adults, with 49% stating that they have bought a ticket within the last 12 months. The next most popular behavior was visiting a casino, and only 26% of adults have done that in the last 12 months (probably much less now).

If You Play, How Do You Win?

For the Powerball, the odds are calculated from two sets of numbers: one from the five white balls (69 possible numbers) and the second from the red Powerball (26 possible numbers). If no one picks all 6 numbers, the jackpot rolls.

People tend to purchase Powerball tickets as the value rises (more rolls from week to week), which is why eventually, a jackpot has to be paid out as more people pick more numbers.

Taking us back to algebra, the probability of winning becomes a simple factorial:

So the probability of choosing all five numbers correctly (white)+ the Powerball (red) number is:

In total, the probability table looks like this:

A low probability of winning.

Mid-Coronavirus

In late March, the Powerball group announced that the minimum jackpots were being cut in half to \$20M, from the previous \$40M. The inital game plan was to slow the growth rate of the winnings, with bi-weekly increases of \$2M, rather than \$10M. They modified that, basing each game on “sales and interest rates” as of April 2nd 2020.

Tickets remain \$2 to play.

Why are they doing this?

1. Drop in Sales
2. Declining interest rates

Even before this, the rules changed back in 2017 to make the game ‘harder’. The thought process was that if the game was harder, and the jackpot got bigger, more people would play. Prices increased to \$2 per ticket, and odds decreased from 1 in 175,223,510 (59 white ball, 35 red ball choices) to 1 in 292,201,338 (69 white and 26 red).

So with the new COVID changes, how has the game evolved? The number of players increased for the Powerball by 14% over the past week, primarily because the jackpot has increased to \$43M according to Lotto Report.

As you can see in the graph below, the number of players dip when there is a big payout, and begin to increase as the jackpot grows larger.

Just for fun, the correlation between the players and prize is 83.2% for Powerball and 40% for MegaMillions, in absolute terms. This is for YTD 2020.

The more people that play, the higher the likelihood that a larger number of tickets will share a Jackpot. Below, N represents the number of tickets with the winning combination (top green row) and the column represents the number of tickets in play. As you can see, the lower the number of tickets in play, the higher the probability that no one has a winning ticket. If only 100M tickets are in play, then there is a 71% chance that no one has the winning ticket, as shown in the below table.

Durango Bill created the equation to build out the following table:

Probability = [COMBIN(N,K) x (P(Win)^K) x (P(Not Win)^(N-K))]

Where N = number of tickets in play and K = number of ticket holders with winning combination. The probability of winning is 1 / 292,201,338 and the probability of not winning is (1 – P(Win)]

So right now, the winning Powerball Jackpot is ~\$40M. If we use the same growth rate from the previous week (14%) that puts the number of ticket holders at 10,233,231. Adjusting the above table for a smaller number of ticket holders reveals:

The probability that no one has a winning ticket is 96.5%. The probability that at least one person has a winning ticket is 3.3%. The expected payout then becomes [(1-.965) x \$40M] which is \$1.38M, with an average jackpot payout of \$0.13 per person.

However, as the number of players increases and the payout increases, the Expected Value increases as well. Players are best off if the jackpot is high and the number of players are low, but as demonstrated earlier, that is not the nature of the game.

For example, our 10M player game, if a buying frenzy happens, and the number of players reaches 35M, like it did back in January, the Expected Value of the game declines significantly across the board.

How do current lotto numbers compare to a year ago, especially with the decline in prize money? MegaMillions shows some discrepancy. The prize package was \$43M larger in 2019 (\$229M vs the current \$186M) and the Sales were ~\$9M different. The sales of 2019 have outpaced the sales of 2020 for most of 2019, with a much larger jackpot (which makes sense).

What is pretty interesting is that despite the 2020 Jackpot sometimes being greater than the respective 2019 Jackpot year / year, sales were still much less. In fact, beginning around Feburary 15th, 2020 sales have been significantly less than 2019. That was partially due to someone winning the Jackpot, but even as the 2020 Jackpot grew, sales remained stagnant. It’s the same story with the Powerball.

It appears that Covid-19 might have an impact on lotto sales. However, it also makes sense – people are in isolation, not going out, and potentially allocating that money towards other resources.

I ran a brief correlation between absolute CCSA (Continued Claims for Unemployment) and MegaMillion Sales. There wasn’t a strong relationship, probably because the lottery is so tilted towards larger jackpots, which skews the impact that any variables might have in this basic analysis. The red triangle shows the most recent point. (A more in-depth look might make a difference.)

Conclusion: Probability vs Possibility

It appears that people are not playing the lottery as much as they used to. I don’t know if that is a bad or a good thing. But the difference between probability and possibility is important to remember.

There is always a 50/50 possibility of something happening – it either happens, or it doesn’t. But the probability of it happening is much lower, especially in the case of the lottery at 1 / 292,201,338.

Stay safe out there everyone!

Disclaimer: This is not written in support of gambling, and I do not support gambling. None of this is investment advice, and it is simply analysis about the lottery. If you feel like you might have a gambling problem, please call The National Problem Gambling Helpline at 1-800-522-4700.