The buzz may be over, but let’s take a look at what your odds were of winning the largest Powerball prize drawing to date anyway. The estimated jackpot was $1.5 Billion, which is more than the GDP of Djibouti (and 25 other countries). The problem with playing in a lottery such as this, as many have pointed out, is that the probability of winning is exceedingly small. By one estimate, it is the equivalent to the probability of being struck by lightning while being eaten by a shark. While that may be an exaggeration, just how unlikely was it to win this drawing? Combining probabilities with payouts gives us a number called “expectancy,” which is a number that we at SIG use frequently to make decisions. In order to determine if a proposition is a favorable one, we have to see if the expectancy is greater than the cost. If it is, and we are risk-neutral, we should take the proposed bet.
So let’s start by looking at the probability portion of our expectancy. In order to win the grand prize, you needed to correctly guess the 5 white balls selected out of 69 balls, and the one red ball selected out of 26. For the white balls, this number of combinations is “69 choose 5,” or (69 x 68 x 67 x 66 x 65) / (5 x 4 x 3 x 2 x 1). This comes out to 11,238,513. Multiplying this by 26 gives the total number of combinations including the red ball, leading to a final probability of having won the grand prize of 1 / 292,201,338. There were other ways to win, as well. For example, if you had all 5 white balls correct but did not match the red ball; the probability is about 1 / 11,688,804. Combining all of the payouts, the total odds of winning something on a single play is about 1 / 25, or 4%.
We are halfway to figuring out our expectancy, but things are about to get much harder. Most of the payouts were set; for example, anyone who matched 5 white balls and not the red would get $1 million no matter how many other people did the same. We can therefore account for a part of our expectancy in playing by summing all of the fixed payouts, multiplied by their respective probabilities. This gives an expectancy of $0.32 for each ticket played (before taxes). Since you have to pay taxes on your winning, for the sake of argument, we will use the highest US marginal federal tax rate of 39.6% to calculate how much your actual bet was worth. This takes this $0.32 down to a value of $0.19.
And how about on that $1.5 Billion? First of all, it is not really $1.5 billion. That would be the total amount paid out if you took the 30-year annuity (paying approximately $50 million per year). If you want all of the money up front, the actual payout would be $930 million. (This still puts you ahead of Grenada’s GDP, as well as 15 other countries.) Of course, the government gets their piece again, knocking this amount down to $561.7 million. So it looks like you had about a 1 / 292 million chance of $561 million, right?
Not so fast. Before playing, you know that the top prize would be split by all of the people who had the correct numbers. Since the probability of winning is so small, there should only be one winner (if any) in any given week, right? The problem was that this large prize amount attracted many, many players. If 50 cents of every dollar goes to the prize pool, we can assume that the prize growing from $500 million to $1.5 billion meant that almost $2 billion in tickets were sold, or 1 billion tickets at $2 apiece. The probability that there was a winner in this group was 99.9999…%. If each of these billion tickets was evenly distributed among all 292+ million combinations of tickets, we would expect there to be, on average, 3 or 4 winners of the grand prize, (note, there were 3 winners) each sharing in the $561.7 million after-tax dollars, so each would walk home with about $164 million. This means the expectancy for the money is our $0.19 from the fixed payouts plus $164 million / 292 million, or about $0.75 for every $2 spent on the lottery. While it looked like this large payout finally made it mathematically correct to play Powerball, each $2 played still loses $1.25, on average.
In the grand scheme of things, it could have been worth $1.25 in entertainment value to know that you could have won the largest payout ever in lottery history. Just don’t think it was a winning bet.