Should you play Powerball? The three levels of naivety (plus one bonus level!)

POST STATUS: Lot of calculations below. Still early days in the checking process. Could be errors. Caveat emptor!

—

This Thursday’s Powerball has a $120 million Division 1 prize pool. It’s huge – the second biggest Division One prize in history. Over 175 million tickets will be sold, at a total cost of $213 million, equivalent to seven tickets for each living Australian.

But should we play? The answer might surprise you. There are three main levels of Powerball naivety, and then one bonus level.

DADS AND ODDS

I care about Powerball mostly because my dad plays it now and then. He’s an educated man with a successful career behind him, but likes the idea of a big prize.

The first thing to know is the odds of winning Division One with a single entry are small. Really really small, as the next chart shows. Division Nine, by contrast, is up for grabs.

Powerball is owned by Tabcorp and each week they lavish a lot of attention on the Division One prize, but it is not given away each week. Far from it.

DIVISION ONE DROUGHT

The last six draws have all gone by without a Division One winner.  Such streaks are not uncommon. Around three-in-four draws end without anyone winning Division One and so the big prize is won roughly once a month:

Division One was won:

• 10 times in 2019,
• 14 times in 2020, and
• 13 times since the start of 2021.

The next chart shows grey lines when the Division One prize was not won, and jackpotted to the next week.

This matters, because the prizes underneath Division 1 are nothing to write home about. Division 9, which you have a one-in-66 chance of winning, averages just a smidge over $10. As the next chart shows, even Division 2, which you have just one chance in seven million of winning, regularly pays out a middling $100,000. Often even less. Most of these other prizes are simply not worth getting excited about.

It’s all about Division One.

The jackpot structure of Powerball makes each week different. Which got me thinking. Maybe, just maybe, Powerball could sometimes have positive expected value. Right? They do a bunch of draws where the Division 1 prize doesn’t go off. Those must have been very profitable. Perhaps that’s how they make their money? And if the prize builds up high enough, to a high jackpot, those weeks could have positive expected value for a player. Maybe, if you played only in certain circumstances, you can expect to come out ahead, on average?

EASY VICTORIES: KNOW YOUR EV.

We need here a very brief introduction to  the concept of Expected Value, which I will call EV. It’s a tremendously useful concept, and a simple one: Your expected payoff is the chance of winning multiplied by the payoff. And the expected value is the expected payoff minus the cost.

For example, if you give me a $2 prize for calling a coin toss correctly my expected payoff is $1. If you charge me $1 a try, my expected value is zero. If you lift the prize to $2.10 , my expected value turns positive.

Knowing EVs is really powerful: They tell you what to do. If an EV is positive, you should play that game or do that thing as much as possible. If they are negative, you should run away.

You have a negative EV if you play roulette at the casino. At Roulette, the combination of the chance of winning and the prizes are designed to give the player a negative EV and the casino a positive EV. It’s a great game for them, in the long run, and a terrible game for us. If you repeatedly play games with negative EVs you will end up losing big.

The EV concept is applicable to real-life situations too, not just games. Investing well depends on estimating EVs, for example, and different postgraduate degrees might have different EVs too. You can extend expected value to any domain where chance or uncertainty is present.

For us, the EV is the main thing we need to know to decide whether to play Powerball. (Assuming, for now, you don’t just love the anticipation of holding a ticket in your hand and setting your mind free to dream).

GETTING RICH ON THE LOTTERY?

How do we figure out the expected payoff of a Powerball draw?  Let’s look at the smallest draw this year, with a $3 million Division One prize. For a rough estimate we can simply multiply the odds of winning each division by the prize. The next chart shows the results of that calculation.

As you can see, Division 9 is doing most of the heavy lifting here. Your one-in-66 chance of getting back around $10.20 is worth a little over 15 cents. The one-in-134 million chance of winning $3 million is worth only 2.2 cents!

After you subtract the $1.21 cost of entry the EV is very very negative. You definitely shouldn’t play for such a small jackpot.

WHAT ABOUT THE BIG PRIZES?

With a bigger Division 1 Prize, maybe your EV goes up? Sure does. Once again we use a simplified technique – multiplying the chance of winning by the advertised prize. Lo and behold, the higher the prize the bigger the expected payoff.

This is a potentially exciting graph if you’re not careful.

Because the next draw, this Thursday, has a Division One prize of $120 million. That would seem to change things.  A one-in-134-million chance of winning $120 million is worth 89.5 cents for Division One alone.  The lower divisions are consistently worth about 40 cents of expected payoff. Add those to the 89.5 cents and you have an expected payoff of $1.30 –  higher than the cost of the ticket! The EV looks to be positive.

“IS IT WORTH PLAYING THIS WEEK?!”

A few years ago I found myself asking this exact question about a Powerball draw with a big Division One prize. My back of the envelope calculation told me the EV was positive and I should play.

What I soon learned is if multiple people win Division 1, they split the prize.  Your odds of getting a winning ticket are the same, but if you share it, your payoff is split. As the next chart shows, Division One has gone off 34 times in the last few years, and it has been shared between multiple winners on 4 occasions.

Here’s the problem:

The number of tickets sold rises as the Division One jackpot rises. And the higher the number of tickets sold, the more chance of multiple division one winners.

You can see the effect in the chart above – three of the seven largest Division One prizes had multiple winners. So the simple calculation I was using above – chance of winning multiplied by advertised prize – is misleading. It’s wrong.

It doesn’t take into account the risk someone else out there has the same winning numbers, is there for your giant cheque ceremony, steals your limelight, splits your pot and rains on your parade.

Check out the $150 million prize in 2019. You think you’re playing for $150 million – the good people at Powerball HQ encourage you to think that – but as lightning strikes and the one-in-134-million chance you have the winning numbers becomes reality, you also discover your lusted-after prize is actually a mere $50 million. (Sure, $50 million is nice, but it’s no $150 million, is it?)

THE THRILL

This is the point that got me really excited: I wanted to quantify this. If there is a relationship between advertised Division One prize and the number of tickets sold, and a relationship between tickets sold and the odds of multiple winners, could we therefore calculate a more realistic EV for each draw? I wanted to do this ever since the $150 million jackpot back in 2019, but it was only this year – once I learned to use R – that doing so became possible.

I scraped data from the internet that let me calculate the number of entries in every Powerball draw in the last 20 years. Here’s data from the last four years, showing that entries rise dramatically when the Division 1 jackpot rises. The red line is a model of a linear relationship.

Is that red line satisfying? Not really. Might the relationship be non-linear? Maybe! The yellow line in the next graph is a smoothed version of the points.

Your chances of splitting Division One go up and up the more people enter. And that changes everything.

THE THREE LEVELS OF NAIVETE

There’s three main levels of Powerball naivety here.

1. One is not realising the pot can be split if multiple people have the winning numbers.
2. The second is not realising the positive relationship between tickets sold and Div 1 prize: Just because multiple winners are rare, doesn’t mean they won’t happen for the big prizes!
3. The third one is not recognising that the relationship is non-linear, so your chance of splitting the prize is rising most dramatically when the jackpot is highest .  

This latter point is important. Powerball draws with Division One prizes below $110 million can’t possibly have a positive EV. The odds are too bad and the prizes are too low. But draws with high Division 1 prizes – such as this week – could in theory have positive EV. The bend in that yellow line may be helping prevent that theory from becoming reality.

DOING THE MATHS

The maths here were tricker than I first realised – I only solved it after cracking out the Poisson distribution. The odds of winning Division One are low. But enough Australians buy Powerball tickets that when the prize gets big, there’s a strong chance of multiple winners.

This graph shows exactly that. Pay attention to the falling white line –  that’s the odds of nobody winning the jackpot, and it dips below 50% when the jackpot is around 80 million. When the Division One Prize is just over 100 million, the odds of there being one – and only one – winner peak. After that they are falling. Somewhere north of $150 million the odds of there being 2 winners will be greater than the odds of 1 winner.

This chart also reveals that the $150 million Division One prize pool from 2019 that was split by three people was actually a little unusual. The odds of there being three or more winners that day was only a little over 20 per cent. It was actually more likely to jackpot than be split by three people. Just a good reminder that the odds aren’t everything!

Just for fun, here’s the odds of multiple winners in Division 2. I include this graph because it shows even more clearly how the chance of a division having a small number of winners dissipates when the prize gets high. (And also because it looks cool and swoopy.)

Your odds of getting the winning number don’t change when there are multiple entries. But your expected payoff from getting the winning number does change.

As more people enter the lottery, your EV must fall. This is not the kind of lottery where you can trust your instinct (and this is perhaps the most important reason for this blog post – to help people overcome that instinct!).

FATAL INSTINCT

The way entries rise with the Div One prize pool suggests some people may be taking that instinctive, naïve view. The higher the jackpot, the more this matters – the more the naïve view takes you further and further from the reality. This week, for example, with the high prize, knowing you may split the pot is more important than ever. The expected number of winners is 1.3, meaning a hefty chance someone else will have their mitts on your novelty cheque.

I set out to calculate the true EVs given the odds of sharing the prize.

I spent ages hoping the highest EV would be for a Division 1 prize that was not a record high, rather than the highest one. I wondered if there was a tipping point where so many people entered the lottery that despite the higher jackpot you had extra reason to stay away. Perhaps the sweet spot with the least-negative EV would be $120m or so. That would have been a cool, counter-intuitive result!

But no, the dominant factor in the EV remains the Division One prize (at least at the prize levels we have seen  offered so far). As you can see in the next graph, the $150 million prize back in 2019 (orange bar) is the one with the highest expected value.

Notice something? Even accounting for the chance of multiple winners, that $150 million draw had positive EV!

So here’s the final bonus level of Powerball naivete: scorning lotteries altogether. It’s a good reminder to keep an open mind. If this week’s Powerball jackpots (probability of there being no winner is ~ 27%) next week could (maybe, just, barely) be worth playing.

HOW MUCH DIFFERENCE DOES IT MAKE TO ACCOUNT FOR MULTIPLE WINNERS?

At the lower jackpot levels, not much. The gap between the naïve EV (simply multiplying the odds by the advertised prize) and the true EV is highest for the bigger jackpots. Which may be why many people respond so strongly to higher jackpots – they are using a naïve estimate of their EV, failing to realise – as I once did – that the prize can be split.

This next chart shows how the naive EV and true EV diverge:

For this Thursday’s draw, we can expect 175,000,000 entries, some of whom may naively suspect their EV is higher than the ticket price. But actually it is lower, because the expected number of Division One winners is 1.3. That eats away at the Division One prize you can expect to take home.

WHO WANTS A JACKPOT?

The lottery company really wants the prize to jackpot. Not just because their profit on this week’s Powerball draw is $120 million higher if the prize does not go off! After all, they do need to send that prize on to next week. What they crave is the free word of mouth that comes with an even bigger jackpot next week.

I guarantee there will be Powerball mania next week if this week’s draw jackpots and they can offer the biggest prize in history. It will be on the news, people will be whispering about it at the watercooler, and newsagents will have their first reason to smile in years.

I also hope this Thursday’s Powerball jackpots. Because then we can collect more data on the number of tickets sold when there is a $150 million+ prize! There has only been one in history and I’d like a bigger sample.

Could it jackpot twice more? The chance of a $150 million prize jackpotting is so low it hasn’t happened before. There have only ever been three draws with a Division One prize over $100 million and they have all been won. We don’t even know what the prize level above $150 million would be. $200 million? $180 million? Something else?!

WHAT YOU SHOULD DO NOW:

I remember one drunken night at a Chinese restaurant being cajoled to join a Powerball syndicate for a huge jackpot. The memories of the details are dim but the memory of the excitement is vivid. We didn’t win though. I think people get genuinely revved up when powerball ticks over $100m, and they go buy tickets.

If your mates are trying to convince you to join them in buying a ticket this week – and you don’t want to waste your money – please feel free to send them this post. “If we all hold off buying this week,” tell them, “the chance of a jackpot is higher! Let’s wait for next week…”

—

POSTSCRIPT. All this was done in R. R is the major source of my nightmares a free open-source coding language for doing statistics and making graphs (and a few other tricks).

I’m what they call “self-taught.” That term may sound cool but what it means is a huge number of people taught me, rather than just one. I owe massive debts to generous people whose advice and counsel I found on Stack Overflow, Twitter, Runapp, Youtube channels and blogs. Thank you. Errors are mine.

Code is posted on my Github. I’m still in beginner mode and very open to feedback on the code – please feel free to take a look and add yourself to the huge group of people who’ve helped me learn .