Yeah, me too. I just got back and read this thread today and was wondering what the hell happened in it. Heh.
You could both be correct or incorrect depending how the lottery numbers are drawn. If there’s only one set of 49 balls, feldman is correct. After you draw the first ball, there are only 48 balls left to choose from, not 49, then 47, 46, 45, 44- it has nothing to do with the order in which the numbers are chosen.
If there are six sets of balls, and each number is chosen from a different set, then it’s 49^6, the familiar 13.38 million number. Most lotteries use multiple sets of balls, AFAIK.
If order were important, you’d multiply either of the numbers (depending how the lottery was run) by the number of possible orders you could put a given six numbers in. Or 6!=720. So it would be (49^6)x6! or about 9.97e12.
Doesn’t getting the powerball right multiply non-jackpot winnings? So if you win a Pick 3, you’d win X dollars, but if one of the three was the power ball, you’d win 2X dollars or something like that? How much money would you make from all the partial winners?
No, it doesn’t matter what the prize is. The question was if it’s reasonable to buy a ticket. If you buy one ticket in one lottery, your odds of winning are unaffected by the prize amount, so whether it’s “reasonable” or not depends solely on whether you want to waste, er, risk, the buck.
I’ve had this argument with friends who say they’ll only buy a lottery ticket when the prize is over $14 million (ok, $13,383,816 if you’re a stickler). Sure, that would be a reasonable strategy if you were able to play the lottery more than 13,383,816 times. Then your overall return on investment expectation would be positive. But in the real world, buying one ticket in every lottery that has a prize over $13,383,816 isn’t any more “reasonable” than just buying one ticket in every lottery. In both cases, your odds of winning sometime before you die are vanishingly small. And, in fact, if you only play the lotteries that have a prize over $13,383,816, you’d likely be playing in fewer lotteries, so your real-world chances of winning before you die are smaller than if you just played every lottery.
Even just buying a single ticket, your expectation is still positive (assuming that you won’t have to share the prize). Expectation value is not the same thing as most likely value. Your most likely result is that you’ll be out a buck, but the prize is large enough that even with your small chance to win, it’s enough to offset your very likely loss.
And 49[sup]6[/sup] will not give you the familiar 13.38 million number. In fact, 49[sup]6[/sup] = 13.84 billion. That’s the number of combinations if all draws are independant and order matters. Likewise, 4948474645*44 (this could also be expressed as 49!/43!, or just over 10 billion, is the number of combinations if the balls are drawn without replacement and order matters. That is to say, using that formula would treat the draw 3, 17, 22, 35, 38, 42 as different from 3, 17, 22, 35, 42, 38. In most lotteries, however, order doesn’t matter, so you need to take that 10 billion and divide it by the number of ways to order 6 numbers, 6! = 720. In other words, the number of combinations is 49!/((49 - 6)!*6!), as others have said, which is 13983816, again as others have said.
Can you explain how this concept of “expectation” has any meaning in a real-world situation of buying a single ticket in a single lottery? I think I understand expectation as a theoretical concept, but I just don’t see that it has any actual meaning in this real-world situation. If I buy one ticket in one lottery, my odds of winning have no relation to the amount of the prize, nor to the cost of the ticket. So where does “expectation” come into play? What does it mean?
The odds of winning a given prize multiplied by the value of that prize contribute to the contingent value of the ticket. If the lotto is above 14M$, the contengent value of the tickets is greater than 1$. So, if you spend 1$ to buy something that’s “worth” more than 1$… hey, you’re being a savvy consumer, right? It’s a bargain!
Ok, but again: I understand this as a theoretical concept that has meaning in a large number of trials. But how does it have any meaning in a single trial?
I mean, bringing the OP back into this: if I were going to buy 14 million tickets to cover all possible combinations, it’s obvious that that only makes sense if the prize is greater than 14 million dollars. Spending $14 million to win less than $14 million is clearly a losing proposition.
But how is buying one ticket in a lottery where the odds of winning are 14 million to one and the prize is two million dollars any different than buying one ticket in a lottery where the odds of winning are 14 million to one and the prize is twenty million dollars? Realistically, how are those two cases different?
I understand that if I win, the second case means that I win more money. But in both cases I’m risking one dollar at 14 million to one odds. Same risk, same odds, I just don’t get how the concept of “expectation” or the “worth” of the ticket is meaningful in a single trial. Honestly, I’m not trying to be difficult here, and I do believe that I understand those concepts, I just don’t see how those concepts have meaning in a single trial.
Where is any of this math coming from?
look up, sees Chronos’s response Well, it seems the matter has been put straight.
Well, what if the group raised its odds of winning even more, let’s say they each bought 2000 tickets each and played only when there was a 20 million jackpot or more, if they win, the split is $40,000 or so each. This might seem like a small amount of money but over time and multiple drawings they stand to win more at much better odds than any single player alone. It is spread out, but I for one wouldn’t balk at 40 thou, especially if I know I have a reasonable chance of winning that every week. It seems like it would be comparable to the stock market if not more profitable.
I think it’s clear that the payoff is relevant, even if you buy only one ticket.
For example, would you buy a one dollar lottery ticket which gives you a one in 14 million chance of winning two dollars? I doubt most would.
Having said that, it seems to be a question of how large the jackpot must be in order to justify spending one dollar so that you’ll have a one in 14 million chance of winning that jackpot.
For some people, I’m sure $1,000,000 is a sufficiently large jackpot to convince them to play. I’m also sure there are still others who will refuse to play until the jackpot breaks the $14,000,000 mark. Furthermore, I’m confident there are some who feel no jackpot, no matter how large, justifies the spending of a dollar. Ultimately (and obviously) it’s a personal decision.
Now, let’s say I do decide to purchase a single ticket (for simplicity, let’s assume there’s a single prize (the jackpot) and no smaller “consolation” prizes). How much is this single ticket worth? Quite clearly, it is (or will be) worth all (of the jackpot) or nothing.
However, before the drawing, the knowledge of it being worth “all or nothing” isn’t particularly useful–I already knew that going in. I would claim we need a better concept of the “value” of the ticket.
A natural approach is to consider the average value of such a ticket (with some fixed jackpot amount). Now, I realize we are only talking about a single ticket here, but I claim it still makes sense to speak of the average value of such a ticket.
For an analogy, consider weather forecasts. What does it mean when, on a single, particular day, we say there is a 50% chance of rain? Clearly, either it will rain, or it will not rain, so what does the 50% mean? It means that, out of all the days in the past with similar meteorological conditions, 50% of those had rain, and 50% of those did not. This is what we mean by saying “50% chance of rain” for a particular day.
Similarly, we can speak of the “average worth” (or “expected value”) of a single lottery ticket. We can look at all of the other tickets in the lottery, and see what they are worth, on average. This will provide us with a way to judge the worth of a single lottery ticket.
Clearly, on average, such a ticket will win the jackpot 1 time out of 14 million, and will be worthless the other 13,999,999 times out of 14 million. So we can say that, on average, the ticket is worth (in dollars) the amount of the jackpot divided by 14 million. Specifically, the ticket has an expectation of winning one dollar once the jacpot hits $14,000,000. A $14,000,000 jackpot is the break even point.
Having said all that, I do see your point. There are certainly other factors to be considered. In particular, how able you are to afford $1, along with the effect winning the jackpot will have on your financial situation.
There are many people who can easily afford $1 whose financial lives would be radically changed by winning a “mere” $2,000,000 jackpot. Such people may feel justified in playing the lottery for this reason, even though the jackpot is well under the “break even” point. This is certainly understandable; I’ve played the lottery under the break even point on many occasions, myself.
However, while such issues may ultimately be relevant to the decision of whether or not to play the lottery, they are not relevant to the explanation of “expected value” I just gave. The “expected value” relies only on the cost, probability of winning, and amount of jackpot; it does not rely on affordability of cost or financial impact of winnings. And it is in this sense that we say your expectation of winning the lottery is positive only when the jackpot is bigger than $14,000,000.
Think of it like listening to the weather report and then deciding whether or not you take an umbrella with you when you go out. Actually, that’s not really a very good analogy, but it is a similar kind of thinking.
Now I wished I paid more attention in Math Class.
Nice thought, but wrong.
What if the current prize pot is $14M, and you buy a winning ticket, and 13 other people picked the same numbers? Your prize is $1M, not $14M. And if you’re in a state that pays out over time, you have to compute the net present value (NPV) of the prize, which is substantially lower still.
The amount of money in the pot is simply an upper bound on the winnings, and in most lotteries, it’s the NPV of that amount that sets the upper bound. With that kind of money in there, the odds are that there will be multiple winners.
You can do all the theoretical calculations of the probabilty of winning, the chance of having to share the prize with others, the expected value of each ticket etc. But the value of hope and the “dream factor” can’t really be measured. About twenty years ago I, an earnest young actuarial student, had just completed all of my probability exams and was busily lecturing my brother on the hopelessness of buying Lotto tickets. I told him that his chance of winning was infinitesimal and only marginally better than mine, which was zero since I didn’t have a ticket. And I remember his reply - “Ah yes, but I have so much fun dreaming about spending the money that I’m going to win, that it’s well worth the couple of dollars per week”.
And guess who had the broader smile when he did, in fact, win the first division prize not long after? 
Ok, so if the weather report says there’s a one in 14 million chance of LIGHT rain today, do you take your umbrella or not? If the report the next week says that there’s a one in 14 million chance of HEAVY rain today, does that change your way of thinking on whether to take the umbrella? If the report the next week says that there’s a one in 14 million chance of a major hurricane, do you pack up your belongings and leave town?
I think it’s a great analogy: If the odds of something happening are vanishingly small, does your response change based on the severity of the thing that has a vanishingly small chance of happening? If so, you’d better start building that rocket ship, because there is a non-zero chance that the Earth will be hit by a large asteroid during your lifetime. And if not, how is that any different than only playing the lottery when the prize is larger than the $14 million?
You’re mostly right. We’re saying with a grin that you drop a buck when the prize is $14Million and you have a reasonable expectation of getting that buck back. It’s kind of a joke.
The lottery is like voting. The numbers are huge and what you do won’t matter, but you like to think it does.
I asked this question earlier (admittedly rhetorically, to make a point) but I’ll ask it more directly this time:
I claim that you, as well, change your response as to whether or not you play the lottery (assuming you ever play the lottery) depending on the size of the jackpot:
To reiterate: Would you pay $1 to play a lottery in which you have a one in 14 million chance of winning a two dollar jackpot?
Correct me if I’m wrong, but I’m thinking you wouldn’t.
From here it’s clear there must be some minimum jackpot for which you would be willing to play such a game. $100? $10,000? $100,000?
How large would the jackpot have to be in order for you to play? What is your reasoning behind arriving at this figure?
Note that, whatever your reasoning may be, it is not my intention to claim that your reasoning is incorrect. It’s your own personal decision on whether or not to risk one dollar for a one in 14 million chance at the jackpot; many factors may be involved that I am not aware of.
My own personal (hypothetical) example: Say someone came up to me and offered a lottery ticket at the cost of a single penny. The jackpot is announced as $50,000,000, but the chance of winning that jackpot is one in 1 trillion.
This means that the expected value of a single ticket (assuming the jackpot will not be shared) is 50,000,000 / 1,000,000,000,000 = $0.00005, much smaller than a penny, and mathematically a losing proposition.
Would I play? Hell yes I would! My reasoning is based on two simple facts: 1. A single penny is virtually nothing to me; I’ve tossed pennies into fountains before with no expected return, and 2. Winning $50,000,000 would be an incredible financial boost for me. The benefit (regardless how remote that chance of happening is) definitely outweighs the trivial cost here, for me.
Another example: Another person offers to sell me a lottery ticket, only this time the ticket costs $1,000,000. However, this ticket gives you a 90% chance of winning a $10,000,000 jackpot. Clearly this is a game made for winners.
Would I play? Hell no I wouldn’t! My reasoning is based on one simple fact: 1. I can’t afford $1,000,000 in the first place.
I wanted to give these examples to illustrate a point, but the fact is that this point is really beside the ultimate point, that ultimate point being the meaning or interpretation of “expected value”.
(Note that both of my examples are intentionally constructed to demonstrate that “expected value” by itself is neither a necessary nor sufficient condition on which to base my decision for either playing the lottery or abstaining).
Back to the “ultimate” point: When we talk about the expected value of a lottery ticket, two very important things are stripped out of consideration: 1. Affordability of the ticket, and 2. Financial impact of winnings.
Basing a decision to play the lottery on expected value simply boils down to 1. How much does it cost to play?, vs. 2. What is the expected (average) value of my ticket? The two points mentioned in the previous paragraph are not factors in such a decision.
In this sense I’m agreeing with you–It is foolish to claim “expected value” is the only relevant issue in playing the lottery. However, you were asking about the significance of “expected value” in this sense, and I’ve tried to answer that in this and my preceding post (more explicitly in my preceding post, actually). I’m not sure how successful I was. Is any of this clear or helpful?
I feel like I kind of left that last post dangling, without giving a clear summation of my point about expected value.
The only sense I can think of in which expected value would be the sole factor on which to base your lottery playing decisions is this:
Say you’re a rich miser. Your only goal in life is to accumulate as much money as possible. The idea of tossing a penny into fountain makes your insides cringe–That’s good money right there! On the other hand, winning $50,000,000 will not make a significant financial impact on your life. Sure, it will make you happier, mo’ money is always mo’ better! But ultimately, you’ll just toss the $50,000,000 in the corner of your vault, to go along with the $50,000,000,000 that’s already there.
This is the type of person who would base their decision solely on expected value. The cost of the ticket? Not a problem, he can afford it easily. The amount of the winings? Significant, but only to the extent that it outweighs the cost. If he can spend $1,000,000 to gain a net of a single cent, he’s ready, willing and able to do it.
For this person, the factors of 1. Cost of ticket, and 2. Financial impact of winning, are not factors. I expect such a person would base their decision only on the expected value of the lottery ticket (vs. the cost of the ticket), since that is the factor which ultimately determines whether the game is a winning, or losing, proposition.
Doesn’t the powerball require you to have the powerball number in the powerball spot. Meaning that the 5 regular numbers could be in any order, but the powerball had to match exactly. What would the odds be of winning that.
Cabbage, you made some good points. I’m not going to try to quote your post and reply to specific points because, well, I’m just to lazy to do that to such a long post.
The answer to your first point, though, is that, no I really don’t change my response based on the size of the jackpot. My response to the lottery is that I don’t really play. The last time I bought a ticket was years ago. Why don’t I play? For the simple fact that the odds against winning are 14 million to one. It doesn’t matter to me if the jackpot is a million dollars or a billion dollars, my odds of winning are still so small as to be essentially zero.
And I think, perhaps, you missed the main thing that I don’t understand. There are people in the world who say and apparently truly believe that it is somehow LESS foolish, or less of a waste of money, to buy a lottery ticket when the jackpot is over $14 million than when it’s less than $14 million. And when I ask them why, they lecture me about “expected value”.
What I claim is that in a case such as the lottery, with such long odds, when you are talking about buying a VERY small number of tickets relative to the odds of winning, that “expected value” is a meaningless concept. In the real world, there just is no such thing as the “expected value” of a single ticket in a 14 million to one lottery. There is what I would call an “expectation”. The “expectation” is that you won’t win. With one ticket, or even hundreds of tickets, the rational expectation of the outcome is that you’ll lose.
So using the theoretical/statistical concept of “expected value” in a situation where the expectation is that you’ll just lose, is what’s foolish. Play the lottery or don’t, I don’t care, but don’t try to convince me that buying one or a few lottery tickets only when the prize is over $14 million is somehow “better” than buying a few tickets when the prize is less than $14 million. That argument just doesn’t make any sense. When the number of trials that you’ll be able to participate in is so incredibly small relative to the odds against winning, the expectation that you’ll lose totally overwhelms the concept of “expected value”.
Getting back to your post, you asked, “How large would the jackpot have to be in order for you to play?” And the answer, honestly, is that it doesn’t matter. Because, as I said, I look at the odds and reason that the expectation is that I’ll lose, so the amount that I “might” win just doesn’t enter into the decision in any rational way. I say “rational” because, yes, I have bought lottery tickets in my life. But the decisions to buy were purely emotional; I’ve never looked at the prize amount and said, “OK, now it’s high enough to risk the buck.” I just go, “Eh, I feel like wasting a few bucks on the lottery this week.”
Now, your other points are well-taken, about there being more than one factor in a decision of this kind. But when you say that “It is foolish to claim “expected value” is the only relevant issue in playing the lottery”, I say that it’s foolish to claim that expected value should be ANY part of the decision.
Don’t get me wrong, though, I’m not saying that expected value is a meaningless concept. I’m saying that in the case of a lottery, where the odds of winning are so incredibly large compared to the number of tickets that you can reasonably buy in your lifetme, that using expected value as ANY part of your decision to play or not just doesn’t make any sense.