I agree with Sam Stone and Enderw24: It doesn’t make sense to strictly apply expectation value theory if you are not going to play the game long enough to realize the expectation value.
If it is worth it to someone to forgo $1 of spending each week in order to have a small chance at winning a huge fortune, then I don’t think one can knock it just because the expectation value is that he will have a loss. [And, on the other hand, if the Publisher’s Clearing House sweepstakes, whose only cost is the price of the stamp, has a positive expectation value that does not necessarily mean you should go ahead and play.]
I’ve found that some people who are too wedded to the mathematics of the expected return have a hard time believing this.
As for me, I simply don’t believe that winning some huge jackpot would make me very much happier … so I forgo the lottery.
As a hijack, here is a fun probability theory question: Assume you are offered a game in which there are 100 pieces of paper with numbers on them put into a hat. You have no idea about the distribution of numbers except that they are all unique. You are allowed to pick them out one-at-a-time and either throw the paper away or keep that one and stop playing. If you stop on the highest number in the hat, you win. You don’t win anything if it is not the highest number. If you are offered this game at 4-to-1 odds (i.e., you pay $1 to play and win $4 if you win), should you play the game and can you describe a winning strategy? [For the purposes of this puzzle, let’s assume you can play enough (with different numbers in the hat each time, obviously!) to realize the expectation value…So, the question is whether or not you can put the odds in your favor.]
Not only are the winnings taxed, but the big ones are usually quoted assuming you will take the winnings in equal chunks over many years. You do get an option to collect a lump sum, but for a reduced amount. Thus although you might see a prize of $10m, to collect it at once you might get “only” $5m. Then pay tax at 39% plus whatever state tax and all of a sudden your $10m has become $2.75m (not that I’d turn that down, mind you). That makes the lottery odds even worse than they appear.
And I wonder if my £2 of premium bonds bought for be by my granny in the 50s means that I am breaking US law?
Well, for my husband and I, it’s because our savings come out of our paycheck before we get it. We buy savings bonds with the money because it’s easier to say no to an impulse buy if the money is locked away in a bond, rather than sitting in an account. We have discussed halting the savings program to pay off the bills, but once you’re used to x-amount more coming in, it’s hard to cut back to where you were. We might be tempted not to start saving again.
Another thing I am guilty of. I think American culture accepts that it’s normal to carry debt. We’re a buy-now-pay-later, instant gratification culture. Americans have never really been into self denial. A lot of advertising whispers to consumers that you DESERVE to have these things, so what if you can’t afford them. You can pay them off over time.
We’re also a culture obsessed with status symbols. Why buy designer jeans when a $12.99 pair from a discount store will be just as servicable? Or a sweater whose cost is ten times as much, merely because of the logo emblazoned on the front? Because it’s a form of non-verbal bragging: I can afford to grossly overpay for my clothing. Not only do I have an ineffecient, expensive SUV, but I have one with an interior designed by Eddie Bauer, even though a sedan would serve my needs. Cell phones, jewelry, designer clothing, $200 tennis shoes, artificial nails . . . all designed to shout: “I have disposable income! Envy me!”
Very true. Or, they think that Social Security, or their pension plan will support them. I don’t think the lessons of Enron were learned very well when it comes to pension plans. Just like cancer, everyone thinks it won’t happen to me, not my company.
And I think they fail to realise just how piss-poor of a lifestyle those who depend on Social Security as their only means of income have. I used to deliver Meals-On-Wheels to these people, and they lived with poverty and despair in run-down apartments, trying to decide between food and medicine.
The fact of the matter is, frankly, people don’t want to save money. It’s just too darn fun to spend it.
Um, I’m not sure that qualifies as an axiom. The expected outcome of an event it the sum of all possible outcomes multiplied by their probabilities. So if I flip a coin, heads I get a dollar, tails I get zero dollars, the expected outcome is: (.51)+(.50)=.5 or fifty cents. If the expected value of a bet is greater than zero, then in a strictly mathematical sense, it is a good bet. But expectation is not the only parameter on which to appraise a bet. (I’m bad at grammer, was that last sentence ‘passive voice’?)
Another issue is the variability of the outcome. Consider a coin toss where you either win a dollar or lose a dollar. The expectation is (.51)+(.5-1)=.5-.5=0. You “should” be indifferent about taking the bet. Consider a second coin toss, instead in this case you can either win a million dollars or lose a million dollars. Here the expectation is still zero, but the variance is huge. It would be a fool’s bet, unless that fool was a billionaire.
A third paramater to consider is a person’s taste for risk. This is strictly subjective.
So if expectation is positive, i.e. greater than zero, it still is not necessarily a good bet. That may be the mistake the gentleman with $500 worth of tickets may be making. If the lottery payout is high enough, then the expectation is positive and it is, therefore, a good bet in a strictly mathematical sense. However, since he is still basically guaranteed to lose $500, it is not a wise bet. Then again, maybe he loves risk. Who knows? At least it isn’t as dangerous as tailgating.
Even on weeks when the pot is “small”, buying one ticket isn’t crazy or stupid if you enjoy the opportunity of daydreaming about winning. For the fantasy value of the game it may be worth a buck, even if you never win.
IOW?? Anyway, I think this question was addressed by the variance point made above. A bet with extremely high variance is riskier in the sense that your well-being will be affected to a much larger degree by the outcome than a bet with a low variance. In the case of the lottery, the variance is high in the sense that you may become a millionaire, and low in the sense that you are practically guaranteed to lose a buck.
Yes. To be strictly proper, you would have to estimate the expected number of people to get the same winning number and factor that into your calculations. Whether a lotto ticket is worth all that math is another question altogether…
Yes and yes. It depends first and foremost on you taste for risk. This is often based on personal preference along with where you are in your retirement planning. A young person can afford a more risky investment than someone about to retire. A good financial advisor should be able to help you make your taste for risk plain. Riskier investment opportunities tend to have higher expected (and realized) rates of return to compensate for the risk. IIRC, small stocks have a higher rate of return than large stocks because they are riskier. Of course, they are riskier. You may have noticed that some investment funds, for example, invest in a bunch of small stocks to spread the variability around and reduce risk.
But notice this: even if you are in a position to take on some riskier investments, they will still make up a small part of your overall portfolio. Also note that in the above I used the word “variance” not in the strict mathematical sense of it.
Regarding the OP, I think people are just bad at estimating risk, making those estimates meaningful, discounting the future, and so on. Someone else noted that we’re not evolved for the world we live in, and the human mind isn’t hardwired to handle the real probabilities we face every day. People tailgate and change lanes in traffic jams. Additionally, properly assessing risks and whatnot is expensive in the sense that it takes effort. Maybe the cost of just using a credit card and not worrying about the relative merits of paying it quickly vs. keeping the money in the bank isn’t enough to be worth the effort and stress of managing money. Surely that isn’t a satisfactory explanation, but that’s the best I can do.
Geez, Truth Seeker and KidCharlemagne, I guess you’d heard that one before!?
Truth Seeker’s answer is the easiest one to use to quickly show that you win at 4:1 odds since you will clearly win all times when the highest number is in the 2nd half of those picked and the 2nd highest number is in the first half which is just a tad over 1/4 of the time [(1/2)*(50/99)] for the case of 100 numbers in the hat. You also win in some more complicated scenarios like when the 3rd number is in the first 50 picked and the 1st and 2nd are in the second 50 with the 1st picked before the second…This scenario occurs ~1/16 of the time.
KC is right that the optimal strategy of this type can be proven in the limit when the number of pieces of paper goes to infinity to be that you go through the first 1/e[~37%] of them and then choose the first bigger one after that. Then you win 1/e of the time so you even come out ahead with 3:1 odds. [It’s fun to work this out.]
I think I was told that this strategy can also be proven to be the optimal strategy over all possible other strategies of any type but that the proof of this is much harder.
One has just as good of a chance at being handed $250 million dollars as winning it from powerball. The old “you can’t win if you don’t buy a ticket” doesn’t apply here.
I dont play the lottery, even when the net expected outcome is positive, since the marginal utility, to me, of all the money exceeding $1,000,000 (after taxes) is useless to me.
If I were offered a chance to play a lottery , odds of 1-200,000 in winning $1,000,000, i would buy up to several hundred tickets. If I were offered a 1-2,000,000 chance to win $10,000,000 i would pass. To me, since I don’t need more than a million to live comfortably, all the rest of the money is, in effect, wasted.
In Canada, about 100-150 people a year win lottery jackpots of $2 million or more. So it does happen, even though your personal odds are extraordinarily remote - you have a better chance of winning the lottery than, say, getting struck by lightning or catching the West Nile virus. (By better chances, I don’t mean on a single ticket; I mean being one of the people who plays the lottery, period, most of whom buy more than one ticket. Your odds on one ticket are worse than that.) I don’t believe 50-100 people are being handed $2 million+ cheques for no reason at all unless they buy a ticket.
So I buy tickets. I like dreaming about it. But I don’t see it as an investment.
I’ll probably buy one–just one–ticket for the next drawing and let the computer come up with the numbers. I figure that I’m throwing my money away either way, but at least I should throw my money away when the potential winnings after taxes (assuming single winner) is equal to or greater the odds of buying a ticket. So basically, I only buy when the jackpot is $200 million plus.
There’s an old joke that the response from Vegas casino owners to people who have educated themselves about gambling and have a foolproof system is “Come right in!”
I used to live in McIntosh County, Georgia, where for years the number 1 industry was fleecing Yankee tourists out of their last vacation nickel at games of chance that you’d think any 8 year old could figure out were impossible. (My own family wasn’t involved in these flim-flams, but the local law enforcement officials were totally on the take- if people complained about the cons, they’d be arrested for gambling illegally, but I digress- there’s a non-fiction book that tells the story in detail called PRAYING FOR SHEETROCK by Melissa Fay Greene). I suppose the promise of something for nothing and the notion that “I’m smarter than the rubes around here” combined to make millions for those unburdend with ethics.
When I lived in Floriday there was a man who mortgaged his paid-for house and bought, I kid you not, more than 50,000 lottery tickets during one of the $150 million weekends. He got incredibly lucky and recouped about $10,000 of his “investment” through partial winners; otherwise he’d have been literally out on the street.
Point: people are stupid and greedy.
I just want to pipe in and say that Lottery playing isn’t as simple as people make it out to be and there actually are sub-optimal ways of playing it. The reason being that if more than one person plays the same numbers, the payout is split evenly between them. Thus, the optimal strategy is to pick the numbers with the least number of people sharing them. This becomes quite a psychological exercise.
Some simple rules are:
Avoid a lot of numbers less than 31 since a lot of people use their birthdays.
Those are mutually exclusive. There are a number of Native american casinos around here and a guy I worked with lent me a copy of a pamphlet he bought on how to beat the slot machines. He wanted my opinion of it. I’ve never seen such non-sense. In a foot-note, Epstein, in his book The Theory of Gambling and Statistical Logic** describes one scheme as “hypnotically stupid”, IIRC. This pamphlet was on the same level.
**Good book, BTW. Heavy duty, but good if you know some math.
Unless they actually do, in which case the casino will kick them out if they find them. Such is the case with card counters, but very few others. For instance, the casinos just LOVE system players who have ‘winning strategies’ for craps, Roulette, and baccarat. In fact, they’ll happily give you a pad and paper to work your baccarat system out on. Try that at the blackjack table, and they’ll probably threaten to have you charged under the device laws.
js_africanus: Theory of Gambling and Statistical Logic is a great book, but you’re right - you need a good math background to slog through it. Another book along those lines is “Theory of Blackjack” by Peter Griffin.
You do realize that the money you put in the bank is effectively cancelled out by the credit card debt, and your net worth is the same, don’t you? Having x dollars in the bank and owing x dollars on your credit card is actually worse than having zero dollars in the bank and owing zero dollars.
Good point. I would add the minor point that it’s not necessarily that we are bad at estimating risk, but that other motivations become temporarily stronger than the impetus to avoid risk. Like in your tailgating example, I think we know that it’s dangerous; it’s just that our instinct to win takes over. Humans have a very strong drive to demonstrate our dominance over others, which unfortunately often comes out when we are in our cars, in the form of “I’m not letting anyone get in front of me”.
IIRC, Blackjack is about the only game that can be beaten with skill and modern techniques mean that all but idiot savants and guys with computers strapped to their body can win.
The optimal non-card-counting puts the odds in your favor by a very, very, small amount. My copy of Epstein’s book is buried, so I can’t tell you precisely. It’s something like, at a dollar a hand, you can expect to be up by a buck after 100 hands, or something really insignificant. So it’s essentially 50/50. The optimal strategy is completely mechanical, it’s just a matter of memorizing it. Anybody can do it. If you like gambling, then that is the way to go.
Bear in mind that 50/50 means a ratio not an absolute difference. So after 10 hands you may be down at 7 losses/3 wins, and at 100 hands 65 losses/35 wins. It gets closer to 50/50 as you go on, but the absolute difference can become quite large.
I’m not sure I’m parsing that last message properly, but if you’re asking if it takes a computer or a genius to beat blackjack in a casino, the answer is ‘no’. I could teach you how to count cards well enough to beat the house in a weekend. And if you are only betting $5-$50 per hand, you can do it openly and most casinos will leave you alone. And betting that amount, you could probably make about $5-$10 per hour, with fluctuations from variance of about $100-$300 per hour.
Anyone can do it. All you need to be able to do is add and subtract single digit numbers, keep a running total in your head, and remember a few extra strategy plays.
