The Powerball lottery jackpot has hit a whopping $400 million, and the question on everyone's mind is "At what point should I actually start caring about the Powerball Lottery"? Answering this question is not so simple, because in addition to calculating the odds of winning a potential jackpot, you also have to figure out the odds that you will have to split the potential jackpot.
The expected value of spending $2 on a Powerball ticket is actually A -$1.58 when you ignore the jackpot.
And while the probability that you in fact win the jackpot is outstandingly small a€” 1 in 175,223,516 to be precise a€” the jackpots can grow outstandingly big, which makes it so that at some point, you can actually get a positive expected value.
For instance, when we last looked at Powerball the jackpot had reached a whopping $600 million.
So is there a way we can analyze the Powerball lottery to try to figure out how many people will play the lottery at a given given Jackpot?
Combining this dataset with information from Powerball that shows when a lottery was a winner a€” and how many winners there were a€“ we're able to make this nice plot of lottery participation as a function of Jackpot value.
Please note that following Powerball's shift to a $2 lottery, the minimum jackpot prize a€” what the jackpot resets to after paying out a jackpot a€” has increased from $10 million to $40 million. What we want to find, looking at this data set, is a formula that will help us predict the estimated number of lottery players given a certain jackpot value.

This one assumes that there is an element of growth at an increasing rate, and accounts for roughly 87% of the correlation. We know the probability that the jackpot will be halved, or split three ways, or any other split given the number of tickets bought. And by adding in the fifth, we can get the full expected value for a ticket bought at any advertised jackpot price!
After some analysis of the estimated winnings, each "advertised" annuity jackpot is roughly 1.8 times larger than the cash-up-front value.
The next two charts show the estimated value of both the Annuity style-winnings as well as the cash-up front winnings for both the linear and polynomial models of lottery participation. The expected value for the annuity, however, is break-even when the jackpot exceeds $345 million in the linear participation model and $380 million in the polynomial model. The Powerball jackpot for Saturday’s drawing reached $900 million, a $100 million increase in less than 24 hours, as the record largest lottery payout in U.S. And the odds of a split change all the time, because the bigger the jackpot, the more people buy tickets. If you look at it in the most rudimentary form, your expected value for playing Powerball was not only positive, but also pretty attractive.

That way, we can calculate the expected value at every single lottery jackpot amount in order to figure out when the expected value says it's worth it to play. This chart that shows the probability that there are multiple winners at a given participation rate. The vast majority of Jackpot winnersA prefer to take all the cash up-front, rather than in an annual 30-payment annuity.
So we've decided to really drill down into the math to combine all the factors, including jackpot size, and the odds you will have to split the pot if you do win.
For pre-January 18, 2012 Powerball lotteries, the amount of money collected is the amount of tickets sold, as they cost only $1 prior to that switch. After January 18, 2012, we merely have to divide the take by two to ascertain the number of tickets bought.
The R2A value tells us that this regression line accounts for about 70% of the correlation between the advertised jackpot and number of tickets bought (simplified).

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