Month: March 2022

Thanks for reading this series. Your reward for getting this far is a juicy post full of fun stuff you can use to get an edge in the lottery. I didn’t expect it was possible to play the lottery strategically, but I was wrong.

I’m excited by what I found. Some numbers will give you an edge. I’ve never seen this analysis done before and I’m not sure why. I guess anyone else who does it keeps it to themselves!

My first post explained how we fail to recognise that other people will share our lottery win, and how that ruins everything. In that post I made a simplifying assumption that the numbers people choose were all random. If we all chose random numbers, then the risk of sharing the lottery win is affected only by the number of people who enter the lottery.

But that’s not the case. Yes, the numbers drawn by the system during the Powerball draw are random. But the numbers chosen by players when they buy their tickets? They may not be random. In fact, with online purchasing, choosing your own numbers is easier than before. No need for a pencil. All our superstitions and dumb heuristics are in play here, we just need to get the data to find how they affect our number choices.

Without further ado, let’s get to the good stuff!

Note that in the title of this post I say “edge”, not “chances”. That’s deliberate. I’m not able to change your odds. Your odds of winning are given by the number of balls in the system. But I can help you change something else that matters – the amount you win if your number is drawn.

As we explained in Post One, expected value is your odds multiplied by the amount you can win. This post is about how to improve your expected value. You do so by choosing a Powerball number that other people aren’t choosing. Ones on the left of the graph above. Unpopular numbers. That means you share the prize pool with fewer other winners and take home more lucre. You can’t improve expected value by much, mind you, but you might as well lift it as high as you can!

Think of this like harm minimisation for your lottery addiction. Most of the time, playing the lottery is the financial equivalent of shooting crack cocaine into your eyeball. It’s not financially healthy. In this metaphor, this post is like making sure you’ve got a clean syringe. If I can’t stop you doing it, maybe I can help you find a way to reduce the worst of the harm.

Before we go any further, a little bit of terminology. This post is about the Powerball number. Powerball is the name of the lottery, but also the name of one of the eight balls drawn in the lottery. Seven balls are drawn from a set labelled 1 to 35, then the Powerball is drawn from a different set, numbered 1 to 20. I’m going to show you the right Powerball number to choose. It makes a difference! In a later post we will look at the other seven numbers.

CHANNEL YOUR INNER TODDLER: YOU DON’T WANT TO SHARE

Choosing lonely, unloved numbers matters. Consider this story: A friend of a friend won Division Two in the British lottery. They expected hundreds of thousands of pounds, but only won thousands. They’d chosen their lottery numbers by going diagonally across the slip. That strategy proved to be surprisingly popular, meaning when those numbers came up there were a surprisingly large number of winners, all surprisingly crestfallen.

Let’s look at the key chart again, before we try to infer some lessons about what makes a number popular or unpopular.

WHAT IS SO SPECIAL ABOUT NINE? WHY ARE WE CHOOSING THESE NUMBERS?

Here are the numbers from least popular (1) to most popular (9). What does this says about the rules and biases people have when choosing lottery numbers?

I’m intrigued by the way the most popular four numbers are close to the middle of the range 1 to 20. It’s like people are using an understandable but silly heuristic – trying to minimise the size of their miss. They choose a number close to the middle. That way whether the Powerball is 5 or 15, they can say, Oooh, Close. (But not right in the middle, that doesn’t seem random enough! 10 is not popular.)

Small numbers are also out of favour relative to larger ones. The ten most popular numbers sum to 125 while the least popular ten sum to 75. Do larger numbers seem more “random” by virtue of being less familiar. And do we think more random-seeming numbers are more likely to be chosen by a random process? The human brain is a funny thing.

It’s also surprising to see 13 relatively popular and 8 lower down. This result makes me think people are actually already trying to use a version of this strategy – choosing unpopular numbers and avoiding popular ones. But absent data, Powerball players are all crowding into the same idea and negating its value!

Odd numbers cluster mostly in the top half of the popularity distribution (7 of 10) while even numbers make up 7 of the 10 least popular numbers. Do odd numbers have some appeal, or seem more random? It’s a well-known phenomenon that people will choose 7 if you make them select a number between 1 and 10 at random .

REVEAL YOUR TRICKS! HOW WAS THIS CALCULATED?

The calculation is simple. Compare the number of winners in divisions that need the Powerball to the number of winners in divisions that don’t.

Divisions 2, 4 and 7 don’t require the Powerball to win. They provide our baseline. On any given week, if there are many winners in Divisions that don’t need the Powerball to win, but few winners in the Divisions that do need the Powerball, that might be a signal that the Powerball drawn that week was not a popular choice for players. And vice-versa.

Popular Powerballs create split prizes. For example, last week’s draw. The Powerball was Thirteen(13), a top-four popular choice. Despite a moderate number of entries, the $10 million Division One prize was split two ways, with each winner getting $5 million. Pretty crummy luck for the two winners!

There’s a lot of random variation in number choices (many people get a ticket type called a “quickpick” where the computer spits out random numbers) so we need a good sample size to have confidence in these estimates, but the maths here is not hard: No French polymaths required!

Technical note: Please consider the above graph as a ranking of popularity. I did some normalising and the y-axis does not represent absolute popularity of the numbers.

THE WHOLE NINE YARDS

The chart above is nice because solves a nagging mystery for me: back in 2019, the biggest Powerball Division One prize in history was split three ways. My calculations in Post One argued that week was the only Powerball draw in history with positive expected value, based on the number of entries. The chances of splitting the $150 million Division One prize were modest, I thought, but it was actually split three ways. Three people had the same winning numbers. Perhaps the Powerball that Thursday made a difference? It was Nine(9), the top choice.

If that week the Powerball drawn had been One(1), perhaps Division One would have jackpotted to an even bigger record!

WILL USING THIS STRATEGY PAY OFF?

I got the above numbers by maths. The real-world test is if they are associated with higher prizes. So I checked to see if the bottom three least popular numbers were associated with bigger wins.

The answer appears to be yes! The next graph shows a juicy bonus in Division 9 from choosing the least popular numbers (1,2,16) compared to the most popular ones (7,12,9).

Remember from post one that Division Nine gives you a huge chunk of your expected value. Increasing your prize by even a little bit helps.

The same kind of result is seen for the other Divisions that require the Powerball (with the exception of Division One, because when there are fewer winners in Division One that usually means the number of winners goes to zero and the prize goes down to zero too.)

PARTITIONING

However, here we run into a question of good statistical practice. I formed the hypothesis by exploiting a dataset of Powerball draws between 2018 and now. If I test the hypothesis on the same dataset, there’s a risk I am fooling myself – random variation could be presenting itself in a way that looks like proof!

I need to test the theory on another dataset. Lucky for us, I have all the results from the old Powerball draws between 2013 and 2018. What sort of results do we see if we test the same popular and unpopular numbers?

The result is pleasing. Again the numbers that were unpopular in 2018-2022 are associated with higher prizes.

Does that mean the exact same numbers were popular and unpopular back then? If we run the same analysis of most and least unpopular numbers on that old dataset we can answer the question.

The answer is: not exactly. Some numbers appear more popular in the older dataset, some appear less popular. Nine(9) is a big mover, going from top to bottom half of the popularity charts. Perhaps people change strategy over time? I think a more likely scenario is statistical noise. But importantly there is one consistency between the two lists, and it is the most important fact: The least popular number to choose as Powerball is One (1).

That matters because the least popular option for everyone else is your best option. You want to choose the Powerball the fewest other people have chosen, and the data makes it crystal clear what the best choice is. Choose One(1). Every time.

The next graph shows the size of the increase in prize you can expect if you choose One(1) as your Powerball. (Ignore Division One, it is confounded by the prize not going off. Division Two is also not relevant as you don’t need a Powerball to win it – the result there is statistical noise.) Look at Divisions Three, Five, Six, Eight and Nine. You can expect a higher prize if you chose One (1) as your Powerball. Fewer other winners are in the mix sharing your prize.

Let’s assume you choose to play Powerball next week. (You should NOT ! the prize has gone back down to a miserable $3 million and the expected value is so negative it makes the Russian Stock market look like a good idea.) If you must play, choose One (1) as your Powerball. Few other people will do so, and if it is drawn your winnings will be higher.

But what about in a year’s time? How long will this edge last? Possibly, just possibly, the word will get out. If you share this post with one other person, and they share it with one other person, then over time, it might become common knowledge that the best choice for the Powerball is One (1). And when that happens, everyone will be doing it! And if so the best response will be to change. This is the literal definition of a strategic game – your best move depends on what everyone else is doing!

But remember this: if most people were smart about playing Powerball, there would be no Powerball. The opportunity to get an edge should last for a while yet!

This is post three in my Powerball series. Post One is here, on Naivety. Post Two is here, on Profit. Post Four is coming soon. In that I will look into what numbers are best to choose for the other seven numbers, excluding the Powerball.

Last week Powerball rose to a very rare $120 million Division One jackpot, and I wrote a post about why such jackpots seem so tempting but turn out so frustrating. It turned into one of the most popular posts in the 13-year history of this blog and readers told me they loved it. I was delighted.

If you would like to read it the link is here. The spoiler is big jackpots are not so great because so many people buy tickets. Even if you win, you’ve got a bigger and bigger chance of sharing the jackpot with others.

I published my post on Tuesday and the draw was on Thursday night. It might have been a popular post but it had absolutely zero effect in dissuading ticket sales! In fact, Powerball managed to sell more tickets for last Thursday’s draw than they had ever sold for any draw before. I had a model that said they would sell 175 million tickets. Instead they sold 250 million tickets. The next graph shows how far out of the usual pattern it was.

We had very little data on how many tickets would be sold for such a large jackpot. Last week I was cautious about saying there might be a non-linear relationship between ticket sales and the jackpot size. Today I can say for sure. There is one. Imagine a straight line through those yellow dots – it would go nowhere near the red one. People go absolutely bananas for a big prize.

Because so many people bought tickets (far more tickets than there are unique combinations), two people held the same winning numbers. The prize was split with the two winners each taking home over $60 million. The prize pool was greater than the advertised $120 million because so many people bought tickets and Powerball is obliged by law to give back 60 per cent of takings, so they bumped up the Division One Prize Pool to $126 million.

The amount of revenue from the most recent draw is a record: $300 million. But as you can see in the next chart, the most recent draw was not the most profitable. It is usually the penultimate draw before the jackpot that makes the lottery company the most money.

You can also see from this chart that they do sometimes lose money on the big jackpots. In particular, four consecutive big jackpots in 2020 were money-losers. Ticket sales were no doubt disrupted at that point thanks to the pandemic and national lockdown. However, no draws since early 2020 have been money-losers. We will explain why in a second, but first lets talk about where the profits go.

The profits of Powerball look too good to be true, right? In a way that’s right. Tabcorp does not keep all the money. The government benefits in several ways. For example, state lottery taxes in the state of Victoria take 90 per cent of what is lost (tax is slightly lower if tickets attract GST). That leaves the lottery company with the sliver between the red and green lines . One reason not to feel bad about buying Powerball and losing is you help fund our health system!

The real money-making weeks for Lottery companies (and their regulators!) are the ones just before a jackpot goes off. When the Division One jackpot does not go off, profits rise in a simple, predictable and very steep line, as shown by the triangles in the next graph. Weeks when the Division One Prize goes off are much less profitable.

This is probably why Tabcorp is eager to make lotteries harder to win. The data I’m using for Powerball goes back to 2018. The reason I haven’t gone back further is because prior to that, Powerball had fewer balls to choose and fewer divisions to win. They changed the rules of the game in 2018, creating Division Nine – which was easier to win – while Division One became much harder to win. The intent was creating fewer Division One wins and more jackpots.

Tabcorp is now planning to do the same to Oz Lotto, which is, like Powerball, a jackpotting lottery. Unlike Powerball, Oz Lotto is drawn on Tuesday nights and has a yellow colour scheme instead of purple. They are very different.

“We will also introduce a change to Oz Lotto that is expected to create larger and more frequent jackpots in line with its promise to deliver ‘Big Aussie Fun’,”

Tabcorp boss David Attenborough, speaking last week. (No, not that David Attenborough!)

The changes will make Division 1 “more likely to jackpot (c.40% more combinations)” Tabcorp say in their recent investor presentation. (pdf at link).

Powerball is Tabcorp’s most popular lottery, so attempts to make Oz Lotto more similar make sense for them.

In fact, the popularity of Powerball has been surging over time. This may explain why draws have been so profitable recently – and also why so many people bought tickets last week. There are more powerball players now than previously. Tickets sold each week have risen a lot, and they rose especially during 2020. People were stuck home and bored and had, thanks to the various fiscal supports, money to burn.

The yellow lines go up, but then they seem to stall a bit. Hot demand for low levels of Powerball may have cooled in 2021-22, which is a tantalising explanation for why the jackpot rose so high recently. A climate with fewer ticket buyers lets the jackpot rise.

It’s a paradox that the more people pile in to the low levels of Powerball, (e.g. selling 45,000 tickets for a $20 million Division One prize pool instead of a mere 25,000 tickets) the lower the chance of the prize rising to a level where Powerball might sell 250 million tickets. And make no mistake, they like those long sequences of jackpots. Once they’ve made OzLotto harder to win, Tabcorp might be tempted to come back to Powerball and do the same thing to it.

This is Part Two in my series on Powerball, on Profit. Part One is here, on Naivety.

Part Three is about Strategy. It is coming soon and concerns the burning question of whether you can do better by choosing numbers other people aren’t choosing (yes you can, and I will demonstrate a way to work out what those numbers are. )