I had fun the last couple of days doing media coverage for Powerball. Here is a roundup of the coverage, mostly for my records.
Guest on “Live at Four” WISC-TV (CBS affiliate in Madison, WI) to talk about Powerball, January 13, 2016 (pictures are below).
Guest on “Central Time” on Wisconsin Public Radio in Madison, WI to talk about Powerball, January 13. 2016.
Guest on newsradio WTMJ 620 in Milwaukee to talk about Powerball odds and strategy, January 12, 2016 and January 13, 2016.
There is a lot of bad information out there about the Powerball (now $1.5B!).
Q: What are the odds of winning?
A: 1 in 292 million. Winning numbers from Powerball are chosen by picking 5 balls from the first 69 balls (numbered 1 to 69) and a 6th ball from a second set of 26 balls (numbered 1 to 26). Each ticket costs $2.
Q: Should I play the lottery?
A: Maybe. You have the same probability of winning (1 in 292 million) any time you play. What is different is the payout. If you must play, play when there is a big payout. You have more competitors when the jackpot is high, but winning is relatively rare and you are still more likely to walk away with more winnings when the jackpot is high (even if you have to share the jackpot with others). You can win $1M if you get the first 5 numbers right, but keep in mind that the $1M does not increase if the jackpot gets bigger.
Bottom line: if you never play the lottery, this is a good time to buy a ticket.
Q: I’ve read that I should buy tickets Pennsylvania. Is Pennsylvania lucky?
A: No. Each ticket is equally likely to win regardless of where it was sold. There are some tax implications based on where you live and where you purchased your tickets. Don’t buy a ticket in New York if you don’t live there or you will have to pay New York income tax if you win.
Q: Will someone win?
A: It depends on how many tickets were purchased. 440M tickets were purchased for the January 9 lottery. Assuming that all tickets are randomly generated and 500M tickets are purchased this time, there is a 82% chance that someone will win. We can break down that 82% further: 31% chance of a single winner, 26% chance of two winners, 15% of 3 winners and 9.4% chance of 4+ winners.
If 600M tickets are purchased, then there is a 87% someone will win.
If 700M tickets are purchased, then there is a 91% someone will win.
If 800M tickets are purchased, then there is a 93.5% someone will win.
If 900M tickets are purchased, then there is a 95% someone will win.
If 1B tickets are purchased, then there is a 97% someone will win.
So while it’s likely that someone will win, there is still a good chance the jackpot will roll over again.
Q: I read that 70% of winning tickets have been computer generated. Does this mean that I’m less likely to win if I pick my own numbers.
A: First of all, if 70% tickets sold are computer generated, then this a meaningless statement. Each ticket is equally likely to win, regardless of how the numbers were picked.
Q: Do you recommend picking the same numbers every time?
A: That doesn’t matter. Each combination of numbers is equally likely to win.
Q: An ABC News analysis of past Powerball winners suggests that 8, 54, 14, 39 and 13 have been drawn the most frequently. Should I select these numbers?
A: Not necessarily. Each combination of numbers is equally likely to win. Of course over time some numbers are drawn more than others because that is what true randomness looks like. However, the picks in the past give no insight into which numbers will be drawn next.
Every lottery ticket is equally likely to win. You are less likely to share the jackpot if you choose numbers larger than 30 because so many people pick numbers based on birthdays. The picture below indicates that while each number is equally likely to be drawn, each number is not equally likely to be selected for a ticket. Choosing numbers greater than 30 will give you a more “unique” ticket that offers a high probability of winning the full jackpot.
Lottery numbers as chosen by lottery players are far from uniformly distributed
Q: A lottery expert said: “The only advice I can really give people is buy as many tickets as you can afford.”
A: While it’s true that your odds are winning strictly go up with each additional ticket you purchase, in general buying lottery tickets is not a good investment. I’d recommend investing in mutual funds or education, or saving for retirement or a rainy day over playing the lottery.
If you are playing the lottery, it’s best to go into a pool with other people. The goal is really to come out ahead, not to win it all. Each ticket has a 1 in 292 million chance of winning, so you can increase your probability of winning by tenfold if you go into a pool with 9 other people and split any winnings amongst yourselves.
Q: What is the cash lump sum payout associated with the Powerball? If I win, should I choose the lump sum payout or the annuitized prize paid over 29 years?
A: The lump sum payout is about $806M (now $903M). The $1.3B (now $1.5B) payout is paid out over 29 years (the first payment is immediate). The tax situation is similar for these two options, so I would recommend the annuitized prize. In either case, you would be in the highest income tax bracket (39.6% for Federal and 7.65% in Wisconsin). This means that you’d come out with about half of your winnings. If you take the lump sum, on average you’d come out $0.54 behind for each $2 ticket you purchase if you live in Wisconsin. And that is an optimistic number since you will probably have to share the prize. So it’s not a good investment. The annuitized payment is a better deal, although the future earnings you would collect will be worth less than the money now.
There are some weird lottery tax rules according to which state you lived in or bought your ticket (see Forbes for more). An article on Arizona Central summarizes financial advice and personal happiness levels after winning the lottery. Spoiler: winning the lottery probably won’t make you happier.
Q: Will you be playing the Powerball?
A: No
(Updated on January 13, 2015)
Q: As the jackpot gets bigger does expected value of ticket get larger or smaller due to an increased possibility of multiple winners?
A: Almost certainly bigger. The increased probability of multiple winners grows slowly with the number of tickets sold. I have a picture below. 440 million tickets were sold when the jackpot was $700M and early reports suggest that around the same number of tickets were sold when the jackpot rose to $1.5B. I made a figure of the conditional expected winnings for someone who wins the jackpot, which takes the possibility of multiple winners into account. I included a few different values of the jackpot for comparison. I don’t take taxes or smaller prizes into account here (the smaller prizes would add a constant to all the numbers here).
Your expected lottery winnings conditioned on you winning.
(Updated on January 14, 2016)
A Chicago area man won the lottery for the second time. The Chicago Tribune reports:
Scott Anetsberger duplicated his $1 million win of nine years ago in the same instant Merry Millionaire game, lottery spokesman Mike Lang said.
Despite long odds, Anetsberger isn’t the first two-time $1 million instant winner. Kimberly Pleticha of Villa Park won $1 million twice in the instant Cash Jackpot game–the first time in August 2010 and the second only six months later in February.
Lottery officials could not instantly compute the odds against multiple winners, but did note there have been a dozen or more two-time Little Lotto winners over the years.
What would the odds of winning the lottery twice would be? Well, it depends on how frequently one plays the lottery.
Winning the Illinois Lottery requires picking six correct numbers, where the numbers range from 1 to 52. The odds of getting all six numbers correct is 1 in 20,358,520. It costs $0.50 to play the lottery, and there are three lotteries per week. Assuming that each lottery is independent (a reasonable assumption), one would have to play the lottery 20,358,520 times, over average, to win (using the geometric distribution). If one plays the lottery three times per week, then it would take 130,500 years to win the lottery once at a cost of more than $10M.
Winning the lottery twice can be modeled as a negative binomial random variable. Assuming that our lottery winner plays the lottery three times per week before and after winning the lottery, then it takes ~261,000 years, on average, to win twice.
Since it is only newsworthy to report additional wins by those who have already won the lottery, then we are really only interested in the odds that a lottery winner would win the lottery again. This is a different question. Assuming that our lottery winner continues to play the lottery three times per week, then the odds of winning again are same as the odds of someone else winning the lottery for the first time: 1 in 20,358,520 per lottery. That is, it would take our lottery winner an additional 130,500 years to win the lottery.
If someone plays the lottery more than three times per week, then the odds of winning go up.
Of course, many people play the lottery, so the odds that someone wins the lottery twice over their lifetime is much, much higher. I tell my students every semester, “Someone will win the lottery. Just not you.” If 130,500 people buy one lottery ticket per game, then there would be a two-time winner every 2 years, on average.
Little Lotto involves picking five correct numbers, where the numbers range from 1 to 39. It is easier to win, but it has a lower payout. The odds of winning are 1 in 575,757, which means that one is 35 times as likely to win the Little Lotto than the regular lottery. It would take 3691 years to win Little Lotto once (by playing three times per week) and 7382 years to win it twice.
Given that there have been 12 two-time winners in Little Lotto in its 23 years of existence, there there is approximately one two-time winner every two years. Given my assumptions, this would suggest that ~3691 people buy a Little Lotto ticket every time. That seems a bit low to me. But I have a head cold and maybe it has temporarily impaired my mathematical abilities.
A seven-time lottery winner’s advice for winning the lottery is to invest more (not less!) of one’s money into buying lottery tickets, as long as one can afford it. He also recommends treating the lottery as a job: the lottery is a skill, and one can improve at it after investing a lot of time. While skill plays a role in playing the lottery (identifying which numbers to pick and identifying which games have the best payoff), I’m pretty sure that this is bad advice. The expected payoff for the lottery is negative, meaning that on average, you are guaranteed to come out behind. The variance in earnings is large, meaning that over many attempts, it is possible that you can come out ahead. But given that one comes out ahead, it would be foolish to attribute one’s success to skill. But maybe I’m missing something.
For the record, I do not recommend gambling or routinely playing the lottery.
For more, read Mike Trick’s post on conditional probabilities and March Madness odds.
Related post:
The same six winning lottery numbers turned up in two consecutive drawings in the Bulgaria lottery earlier in the month (1 chance in 5.2 million). Carl Bialik in the WSJ writes about the odds of this happening. He notes that “With so many numbers colliding each week, the lottery might be the ideal proving ground for something that statisticians have long recognized: Given enough opportunities, the seemingly impossible becomes plausible.” He explores several lottery issues in more detail in the Numbers Guy blog. Statistician David Smith also blogged about the Bulgarian lottery.
Although the lottery is random, the people who play it are not. I had always intuitively known this, but the picture below illustrates this quite nicely. Apparently, people making lottery picks based on birthdays, for example, skews the picks toward smaller numbers.
Lottery numbers as chosen by lottery players are far from random
The lotteries are designed such that the expected winnings are negative when accounting for the price of the ticket, since the probability of winning is so low (E[winnings] = P(win)*Jackpot – Ticket Price). When the jackpot grows large enough, the “average” lottery player can come out ahead (although there really is no one at the average – there are a couple of winners who really skew the average). In March 1992, the Virginia lottery almost guaranteed a true winner. It offered a jackpot of $27M to a single winner whereas it cost $7.5M to purchase all Choose(44, 6) combinations of possible tickets (by piacking six of 44 numbers). Of course, this strategy could backfire if there were many winners. However, a group of 2500 people accepted this challenge and pooled their resources. They ended up being the single winner, and after a legal struggle, they were awarded the jackpot. The Virginia lottery was subsequently changed to be less lucrative.
Do you play the lottery?
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