Think of a lottery as a stupidity tax - the dumber you are, the more you pay. I only have to pay $6 a week.
Your chance of winning goes up tremendously when you buy that first ticket. After that, the law of diminishing returns sets in.
I believe the issue Carryon is getting at, although incorrectly, is that the simpler the numbers pattern (the less words to describe the pattern, for example) the less it can be described as “random”. The less random a sequence, the more likely someone will have chosen it - or multiple someones. The news said they think something like 2/3 of all possible combinations had been played last draw. Your odds of splitting the prize, particularly with a very non-random sequence, are much higher. (But half of $900M is still a jackpot) But yes, any sequence is as likely as any other sequence.
Would also like to point out people like to play dates - so 1-12 are likely played a lot, and 1-31 also more often played. What I did notice when I tried to pick random numbers in lotteries, before I settled on quick-pick - sometimes humans also try too hard to be random; gotta have one from each decade, can’t have a sequence of two numbers in a row. Well, check winning number history - two numbers in a row is common, and one or two decades of numbers not represented in the winning set is quite common, almost inevitable. Let the machine do the random for you.
That’s the site I used today to buy a couple of tickets. They seem to be legit, but it’s my first time buying online and I have no idea how it will all turn out when I win.
IMHO, deciding whether or not to buy a lotto ticket has absolutely nothing to do with expected return and cost/benefit analysis. It’s a trivial amount of money for a chance at a life-changing amount of money, not a long-term gaming strategy. Expected returns should be considered for bets with higher risks and lower payoffs, but the lotto is all about hoping for a miracle.
Sure the average return (return per game) is greater than 1, but to be guaranteed a win, you need to play 300,000,000 games…
The buzz is about the greater than one issue, because under the jackput formula,
while people are buying tickets in a frenzy , they are increasing the jackpot TOO, so the return is always going to be greater than one,… so the more people who buy tickets, the more the ROI reduces toward one, but hey, if you are in this for the long haul, you are guaranteed to make money.
For most short term players, its just a lottery, and you can get most of your investment back if you buy many games, but you are very unlikely to get any great amount more than your investment back, eg the odds are you 95% ROI if you play 200 games, something like that, and at a hundred to one, you make money. The rules are designed to promote individuals playing rather than rich people turning it into a publicity stunt.
The simplest analysis says that the expected value (prize value multiplied by probability of winning) is greater than the purchase price, so you could look at this as an investment with a positive rate of return. But that ignores risk, and this is something a smart investor should be very concerned about. Whereas bonds are a low-risk investment (i.e. a very high degree of certainty regarding outcome), and stocks are a moderate-risk investment (less certainty of outcome, especially over the short term), a Powerball ticket represents about as much risk as you could possibly inject into the situation, i.e. an all-or-nothing outcome. On average you’ll come out ahead, but no single outcome will be average: for any single ticket, you’ll either get it all, or you’ll get nothing.
Some investing risk is smart. If you put your whole retirement savings in low-risk bonds, you’re probably going to retire poor. The smart investor will put a large chunk of their savings in stocks, accepting some risk in order to give a high probability of having a rate of return that assures a comfortable retirement. The risk gets reduced over the long haul: any one year may give shitty stock performance, but over the course of 20/30/40 years, things tend to average out and you are very likely to get an annualized rate of return that’s consistent with history, i.e. somewhere around +7%.
With the Powerball, there’s no “long term” long enough to satisfactorily mitigate the risk. Imagine that instead of investing $20,000 in stocks each year for the next 40 years, you invest $20,000 a year in Powerball tickets for the next 40 years. At $2 a pop, that’s 400,000 tickets, so your net odds of winning is 400K/292M = one in 730.
Suppose a friend said to you, “I have an investment opportunity for you. The buy-in is $400K, the likelihood of returning $1.3B is 1:730, and the likelihood of losing your entire $400K is 729:730. You want in?”. Would you expose $400K to that kind of risk? Or would you rather put it in stocks for a few decades, where you’ve got maybe a 99% chance of breaking even, and a 95% chance of getting a rate of return that beats bonds (YMMV)?
You’re right that everyone ignores risk, including me in my post above. I thought about it but the risk numbers get really silly. I didn’t even want to talk about it because everyone understands that buying one lottery ticket is the riskiest thing in the world. But it’s wrong that you either get it all or you get nothing. You could win a lesser prize.
For the hell of it, I calculated the standard deviation of weekly returns for the S&P 500 (excluding dividends) and this week’s single ticket lottery bet. The S&P 500 had a standard deviation of weekly returns over the last three years of 1.66%. One lottery ticket this week has a standard deviation of 2,357,613%, including the value of lesser prizes. A single ticket in the lottery is therefore 1.4 million times riskier than the stock market. It might be the riskiest investment ever in the history of man.
The classic way to mitigate risk is to diversify. In the stock market, you buy a lot of stocks that don’t all move in price together. In lottery terms that means buying more tickets that aren’t all the same numbers. If you have $300 million lying around, you can perfectly diversify and eliminate the risk of not winning the jackpot. You can’t control the risk of having to share the jackpot.
And even this strategy only works if the lottery odds and the jackpot stay the same. Unfortunately, the jackpot will shrink as soon as someone wins it and then the lottery will go from being a debatably good investment to an indisputably poor one overnight.
Think a little more about what you’re saying here. You’re saying that, just by designating a meaning for a number, a human can influence the outcomes of machines blowing ping-pong balls around. In other words, this means that humans have psychic ability. Is this really what you think?
They change dramatically if it is a new record and gets in the news. I did not hear about it until it got over 800 million.
I walked down to the liquor store and plopped down $20 five 5 quick pick tickets. I also got a six pack of beer. While I was deciding what beer to get 5 other people bought lottery tickets. The guy behind the counter says tickets were selling like mad something that usually does not happen.
Saying that the expected value of purchasing a $2 lottery ticket is $1.06 is like saying the expected value of going to a bar and buying a woman a drink is being able to insert your penis 0.29 inches into her vagina.
I belong to a lottery pool at work. I consider it “rapture insurance”. I don’t think we’re ever going to win, but if we do, I don’t want to be left behind.
I have heard that, in general, the lottery is the poorest return on investment of any type of gambling (which is generally poor overall).
I have been told that illegal numbers rackets will base their games on published lottery numbers (to transparently prove they are fair) and still provide their ‘customers’ with better odds than the legal lottery.
All kidding aside, I’m not buying a ticket. Unless someone actually does buy all 292 million tickets, I think there’s a 4% chance or so that the jackpot will roll over One.More.Time.
On that next round, rolled-over, I expect a jackpot of $2 billion or more. Then I’ll buy a ticket. $2 on 28-38-48-49-56 with Lucky Red 13.
An interesting sidebar to all this is the role of government in it, an institution that normally and in general I tend to support. In years past, governments generally prohibited lotteries and other forms of gambling because they preyed on the poor and stupid and those with gambling addictions. The standard fantasy about winning a lottery invariably focused on the iconic and now defunct Irish Sweepstakes.
Things started to change, IIRC, when Jean Drapeau, the innovative and visionary mayor of Montreal at the time, introduced what I believe was the first lottery in North America in 1968 to help pay for Expo 67 and the newly built Montreal subway. He actually called it a “voluntary tax”. It seemed to take almost no time at all for governments all over to see this as a huge potential revenue source, and suddenly the song changed, and ads burst out all over the place enthusiastically informing the poor and stupid that they could make no better investment than to buy lottery tickets.
Hey, I’m not being judgmental, as I said I occasionally buy one myself just for fun. What amazes me is this hypocritical about-face in government attitude to the exact same scam. I like the Dogbert business model that appeared in one of the Dilbert strips: sell used lottery tickets for half price! Your chances of winning stay basically the same (they round to zero in both cases), but you only spend half as much.
Don’t let the size of the jackpot fool you. The bigger the jackpot, the more tickets they sell this week, and the greater the chances that whoever wins will end up splitting it with several other people. In order to calculate the expected value of a ticket, you need to know how many tickets there are.
Let t= number of tickets. p= size of jackpot. n= number of unique combinations. The expected number of winning tickets then would be t/n, so the actual payout is expected to be p(n/t). And the chance of your ticket winning is 1/n. If c is the cost of the ticket, the expected value of your ticket E = (1/n)§(n/t)-c*(n-1)/n. For large values of n, this is essentially E = p/t-c. If p is $1.4 billion and c is $2, then E>0 if and only if t<700 million.
So the key question is, do you think there will be more than 700 million tickets sold this week? If yes, then don’t buy a ticket.
However, as others have pointed out, having $50,000 may be a thousand times better than having $50, but it’s probably NOT true that having $50,000,000 is a thousand times better than $50,000. The relationship between value and utility is not linear. So really the expected utility of your ticket is even lower than the expected value of your ticket.
OTOH, if you get happiness from dreaming about winning and buying a losing ticket makes you happy for a few days, then that itself might be worth $2.
Using the likelihood of having to share the prize derived from this article in CNN, Could you guarantee yourself a Powerball jackpot?, I revised my expected value calculation for a single ticket. At a $1.4 billion grand prize, and $865 million cash value, one ticket is worth about $1.95 so the answer to the OP is no. By my rough estimate, the grand prize has to climb to about $1.6 billion for a ticket to have an expected value greater than $2. It might do that.
Blue Blistering Barnacle, I was using a financial definition of risk, which measures the difference between expected value ($1.95 or so) and all the possible outcomes (-100% very likely to +80,000,000,000% very unlikely). With my new expected value estimate, the estimated risk would drop.
sbunny8, I think your estimate of the number of winning tickets assumes that they sell a particular set of numbers only once until they have sold all the possible combinations, then they start selling the set again. That’s not true. In the last drawing, 440 million tickets were sold, which is more than the 292 million unique combinations but there were no winners.
The silliest aspect of this lottery - and pretty much all others - is that people deliberate over whether there’s some “logic” that will confirm their decision to buy a ticket or not. Look, when there are millions and millions of people buying tickets and millions and millions of possible winning numbers. you’re kidding yourself if you think you have anything more than a hope in hell of winning. The odds of winning the lottery when buying a ticket are virtually the same as winning it when you don’t buy a ticket. This stuff about which numbers are more likely to come up, how many people are buying, what numbers get played most often, etc. are just ways of entertaining yourself while you watch your overall wealth go down by a couple of dollars. Pul-eaze - don’t delude yourself to taking this so seriously.
No, I didn’t make the assumption you described. I’m aware that “zero winners” is a plausible outcome. But I’m not trying to find the minimum number of winners; I want to estimate the average number of winners. With n=292,000,000 unique combinations and t=440,000,000 million tickets sold, the actual number of winners could be literally any whole number up to 440,000,000. However, the most likely outcomes would be zero, or one, or two, perhaps three. But the average number of winners is t/n = 440/292 (approximately 1.507). Imagine this exact same scenario played out over and over and over. 440 million tickets with 292 million unique combinations. Some weeks, there’s be one winner. Some weeks, it would zero. Some weeks, it would two or three. Once in a while, it would be four or more. But the average number of winners per week would come out to 1.507.
I did oversimplify a little bit. I assumed that all combinations are equally likely to be chosen by players, but the truth is that some combinations get picked more often than others. But we’re working with approximate numbers anyway because we don’t know precisely how many tickets will be sold. I guarantee you that Victor Matheson (the college professor quoted in that CNN article) used an estimate of how many tickets will be sold when he came up with those figures in the article. And I hope you agree that, if twice as many tickets get sold, that would double the expected number of winners, which would cut in half the expected value of each winning ticket.
If the jackpot gets bigger but the number of tickets sold increases proportionately, that doesn’t improve your expected winnings.
The intangible value some people return from the time spent calculating exactly why buying a $2.00 lottery ticket is a bad investment and then posting about it on the internet may roughly equal the intangible value other people return on buying a $2.00 ticket and fantasizing about buying private islands or whatever. Everyone in the former category and the vast majority of the people in the latter category are not going to actually win the lottery.
I’m in the latter category. I’m getting a submarine if I win.
Well, you’re counting how many angels can dance on a pin. The jackpot is at $1.4 billion dollars. Suppose there are 10,000 jackpot winners? Who gives a shit?! Are you actually still gonna mope over winning ‘only’ $140,000 dollars?!?
Given that the five match prize is a flat $1 mil, no matter how many winners, yeah, I’d probably be a bit upset if matching all six paid put at much less.
