Everyone is talking about the record-setting Powerball Lottery jackpot for tonight’s draw. Even in Canada, beyond the reach of official ticket sales, the lottery is making headlines.

I saw this (inaccurate) calculation via social media and decided to use math to answer some of my own questions about the lottery.

**The $650,000,000 Question**

The lottery posts the odds of winning on its website, but I wasn’t sure if the posted odds refer to having *a* winning ticket or *the* winning ticket. In other words, do I have a 1 in 292,201,333 chance of winning the whole jackpot, or is there a chance that I might have to share that jackpot with another winner? Since the difference between winning the whole jackpot and sharing the jackpot is at least $650,000,000, I figure that’s an important question.

5 balls are drawn from a barrel of 69, and then the *Powerball* is drawn from a separate barrel of 26. Once drawn, balls are not returned to the barrel, so are the draw proceeds, the likelihood of drawing a particular number increases. Also, the sequence of the numbers doesn’t matter. In other words, there are several different *permutations* of the winning *combination*. Using my favourite button on the calculator (*x*!)revealed that there are, in fact, 120 possible permutations of a 5-component combination.

So, according to my calculations, the chance of having a winning ticket is:

This matches the odds posted on the website, which tells me that, as narrow as those odds are, they’re just the odds of being *a* winner, not necessarily *the* winner.

**So what are the odds of being the only winner?**

In order to be the only winner, two conditions have to be met:

- I have to have a winning ticket
- No one else can have a winning ticket

While the odds of the first condition are extremely remote (see above), the odds of any given ticket being a loser are almost certain; however, almost certain still isn’t quite certain and I imagine that a lot of tickets will be sold, so a little more math is needed…

Because this probability depends on the number of tickets sold (*n*), which is a quantity that I don’t know, I generated a graph so I could see how the probability changes as the number of tickets increases.

The number of tickets is plotted on the *x* axis and the probability of being the only winner is plotted on the *y* axis.

As you can see, the odds or so low that the number of tickets sold doesn’t make an observable difference on the probability of being the only winner.

**Another scheme**

Now I’m wondering if there another strategy with a better risk vs. reward. What if I guaranteed a share of the jackpot buy purchasing a ticket for every possible outcome? That would cost a little less than $600 Million. If was the only winner, I would make a profit of $700 Million (before tax). Is that possibility worth the risk that I might have to share the jackpot with other winners?

I’ll delve into that question if I have time before tonight’s draw.