# Powerball Probability

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:

1. I have to have a winning ticket
2. 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.

You can zoom in on this version of the graph.

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.