There are games like Pick 3, Pick 4, etc. where each digit is independently chosen. So you could pick 1-1-1. Such games generally have very short times to the next draw, perhaps run twice a day and fixed payouts with no rollovers.
Trying to game the game by by picking numbers others are less likely to pick is problematic. The increase in chances is very tiny. The state will still take its cut (usually 50%) and it takes an ever increasingly large jackpot as they make the odds of winning it smaller and smaller to make it theoretically worthwhile. (I think at one point one of the multi-states games only required a ~$75M jackpot to reach that point. But they have reduced the odds at least twice since then while increasing the ticket cost.)
But even at the theoretically better than break even point you enter St. Petersburg Paradox country where you are vastly more likely to go broke or die before you will win.
Quite right. As I noted, I simplified things just a bit. I hoped that change would simplify the explanation for the OP. Without excessive misleading.
Here’s another way to think of it. If you’re over age 50 you should only buy a lottery ticket on the night of the drawing and only within an hour of when they stop selling tickets. That way your odds of winning are better than your odds of dying from any / all causes before the drawing.
Conversely, if you buy your ticket e.g. the day prior, you’re much more likely to be dead by the time of the drawing than you are to be alive and be the jackpot winner.
The older you get, the closer to the sales cut-off time you have to buy your ticket. By the time you’re late 70s, it’s impossible. The roughly one hour time interval between when they stop selling tickets and they make the drawing is longer than your personal “more likely to win than die” interval.
IIRC Powerball makes those stats available. Or did. And yes, the fraction of players who do random picks serves to disguise, or damp down depending on your perspective, the effect of people who pick birthdays, 1 through 6, etc.
I recall reading about an event where one time the NY (?) lottery had a remarkable number of people pick the same list of seemingly random numbers and that set won the jackpot. So the prize pool had to be divided up between a crazy number of winners. It looked suspicious as hell, and an investigation ensued.
Turns out a local fortune cookie company printed lucky numbers on their fortune slips. Which numbers they picked just once a week than ran off thousands and thousands of cookies all with the same numbers on the slips.
They’d shipped a truckload of matching cookies out to local Chinese restaurants all over NYC. Many cookie eaters played the fortune numbers. Which happened to win. The slips had been printed a couple weeks before the drawing.
I have had two experiences of using similarly spurious logic.
Once I had a ticket in a lotto draw and had left the ticket at home while on holidays. The huge jackpot was won but the winner did not immediately come forward. So I began to tell everyone that, with each passing day as more people checked their tickets, it was becoming more likely that I was the winner. I said that once everyone else had checked their ticket, and lost, then I must be the winner. Unfortunately they publicised where the ticket had been bought and my dreams ended.
The other time provided me with lots of laughs. I, for some unknown reason, had an instant scratch lotto ticket with a prize of $500,000. When I scratched off the first two symbols they were both the winning symbol. So if any one of the remaining symbols was the third winner…
So I carried the ticket around in my wallet with only those two symbols revealed for months. I would pull it out if things got shitty at work and tell people that all had to do was reveal the third symbol and I could take a long holiday. It was amusing to see the number of people amazed at my profligacy. They were horrified that I was just moments away from a big win and wouldn’t reveal it. I was playing this gag on a group of people when one of them pointed out that the tickets came with a limited payout window. If I did have the third symbol and revealed it after the payout window closed it would be a different order of funny. Turned out that he was right and I only had weeks to go. So I completed my scratching and won nothing.
As always probability can be viewed from two POVs: liklihood of a specific outcome of a future event which has not yet occurred versus our state of knowledge about a past event which has already occurred.
Very different intiition and bayesian math applies to those two cases.
Your claim the logic is spurious suggests you’re not accounting for those differences.
Indeed. I was more interested in the range of biases and thoughts punters might have in choosing numbers.
There seem to be divisions in approach. Some people just buy random number tickets. Others give thought to numbers that mean something to them, and there is a small number that believe in patterns of numbers between draws. There is probably an interesting seam of psychology to be mined across such a simple action.
Picking 123456 is an interesting thought process in itself. In picking it, a punter is dismissing the possibility that many other people might have the same idea. Which in itself is an interesting bias.
As I tell others - It seems to me the lottery is a stupidity tax. The dumber you are, the more you have to pay. I only have to pay $6/week.
I used to fill in the card with random numbers - turns out, humans suck at random. In 6/49 (49 balls, pick 6) quite often there are 2 numbers in sequence, whereas humans tend to avoid two numbers in sequence; there’s usually two in the same decade, which I usually avoided too thinking it’s too non-random. I let the computer do random for me now.
A simple exercise will show why people might avoid a sequential ticket - consider all the possible ticket combinations (what, 269 million, or 13.9 million in 6/49) and only one is that particular sequence. The catch is, every other ticket sequence is the same “1 in X”. Most of them just don’t have patterns.
The othe way to look at the odds is - for those crazy Powerball and Megamillions - whatever the odds, 2 to 5 people a year win them, out of 340M. A lot more people than that get hit by lightning, sure, but do 5 people a year die from tipping vending machines?
There’s always prayer. Or the story about Bob, who goes to church to pray that God would let him win the lottery. He prays every day. One day, while he’s praying, a cloud forms… a giant hand appears out of the cloud and points a finger at him. A voice booms out, “Bob, you cheap bastard! You could help me out by buying a ticket once in a while!”
Yes, the chance of any individual ticket winning is absurdly low. On the other hand, SOMEONE will win eventually.
Not true. I play the variable prize games when the prize reaches “pot odds” like a poker player. For example, the chance of winning Powerball is 292,201,337:1. A ticket costs $2. Therefore I buy a ticket only when the top prize exceeds $584 million. My calculation ignores the lesser prizes.
I openly admit that I have lost money on the above system. I used to save my tickets. At the end of the year I would subtract my costs from my earnings. I was never CLOSE to even. IIRC my expenditures were never more than $20 per year. Similarly my biggest win was probably about $20. However just because you lost any specific bet does not mean it was an incorrect play.
You’re certainly right about the idea called “expected value”: That once the ticket:prize ratio exceeds the odds of a win you’re in positive expected value territory. So far so good. But …
Given that you receive no benefit from any taxes paid, a more complete analysis would consider that. You’re going to lose ~40% of the prize money to taxes. So you ought to wait until the prize is a bunch bigger, more like $973M pre-tax = $584M post-tax before you’re actually in positive EV-net-of-inevitable-taxes territory.
Now, don’t forget the headline prize is not the actual amount you win. That advertised number is largely BS: it’s the sum of 30 annuity payments over 30 years. The estimated wins right now as I type are $106M combined annuity value over 30 years and $48.2M actual cash value today.
So to get to actual positive EV territory, you need the cash prize to exceed $973M. Based on the 106/48.2 ratio today, which varies based on long term interest rates, you’d need the headline jackpot to be 2,140M or 2 billion, 140 million dollars. Which it has never done, and probably never will do, at least under the the current game structure.
But the real objection is statistical, not taxes and annuities vs cash and time value of money. And it goes like this:
What is the EV of a $1 bet on tossing heads on a fair coin once? The textbook answer is 50 cents. But you will never have that outcome; it’s completely impossible. Instead you will either win $1 or $0. If the number of trials is small enough, the EV will not and cannot occur.
Now do the same thing, but make 10 bets on 10 tosses. The outcome will be closer to $5.00 than to $10 or $0. You’re starting to see EV at work. Play this game 100 or 1000 times and the EV is extremely predictive. You’ll come out real close to $50 and closer yet to $500.
That’s for a game with 50/50 odds. it only takes a couple dozen trials for the outcome to converge to roughly the EV.
When the odds are ~1:300M, you need to play the game billions and billions of times before the EV starts to emerge from the baseline of loss, after loss, after loss.
Needless to say, you won’t live to play Powerball billions of times. 3 draws per week times 52 weeks times a 100 year lifespan is ~15K games. You’d need to buy millions of chances in each one to have enough trials to see EV emerge.
And you’re especially screwed becuase you only want to play when the EV is positive and the jackpot exceeds $2.14Billion.
So no, you’re not playing smart. You’re just losing less than the other dumber players.
Lest you take all this as an insult, let me say I too buy tickets regularly. Spent $20 for 10 chances at the aforementioned $48M cash prize about 2 hours ago.
The math is unequivocal: it’s a losing bet, a sucker’s bet. I do it anyhow. You evidently do. But in one sense you play smarter than I do since you spend less than I do. So there’s solace in that.
My father plays Powerball every week, as he has for years; I think he buys $10 of tickets (random numbers) for each drawing.
He absolutely knows that the odds of him winning more than a couple of bucks are incredibly slim. He argues, “A few dollars for a ticket is the cost of entry for being able to legitimately dream about what you’d do with a few hundred million dollars.”
@LSLGuy the textbook game in this case is Bernoulli’s: start with a prize of $1, and flip a coin at each stage. The game ends whenever heads appear; the prize is doubled when tails appear.
How much would one pay to enter this game? Considering the expected winnings are “infinite”.
Another way of looking at it: If the choices are merely “pattern” vs. “no pattern”, it’s vastly more likely that “no pattern” will win, because there are a great many different “no pattern” draws, and only a few “pattern” ones. That’s what makes it feel like a pattern is less likely. But that reasoning is only valid for patterns or no-patterns as a whole, not for any individual pattern or no-pattern.
Nah, I’ve got a better deal. My plan for winning the lottery is to find the winning ticket lying on the sidewalk. It’s only a very small difference in the probability of winning, but costs me nothing.
This is going to be off on a tangent, but here is a cute little exercise: I am going to present you with a random string of 0’s and 1’s: 0 0 1 0 0 1 1 1 1 0… What is the expected time to wait (from the beginning) until 10 first appears? It’s 4. But what about 00? The expected waiting time is 6.
Next, consider a random string of length n using, let’s make it the letters A and B. What is the expected number of occurrences of some subword? It is pretty trivial to calculate this, and it does not depend on the subword, only on its length. But! What is the probability that a random string contains, say, at least one AA? Versus the probability that it contains AB?
Do professional poker players take taxes into consideration? I doubt it.
On the other hand, you might win a lesser prize, ranging from thousands down to a free ticket. My gut is that “gain” is insignificant.
Furthermore, back in April, Mega-Millions adjusted their odds but raised their price from $2 to $5. That puts my threshold at about $1.45 BILLION! We will be lucky to see that once per year.
I take no umbrage whether you play or not. I stand by my pseudo-pot odds rule.
