Lotto Probability

[69dodge]

I'm having difficulty still with the concept or context of "utility" as it applies here (as well as EV). I've done some looking around, and actually found a paper written on gambling which involved these terms...but the paper was far too advanced in probability theory for a layman.

Does anyone care to try to flesh out the definition of these terms? I feel there's alot being said here that isn't particularly difficult to understand if you speak the lingo...?

Expected value (equivalently, expectation) is the simpler concept. Basically, it's the average amount you'd make per game if you played the game over and over again forever. (If you'd lose money, it's negative.) In the long run, each of the possible outcomes of the game happens as often as its probability says it should happen. If the probability of winning a single lottery game is one in a million, for example, then in the long run you'll win the jackpot in one game for every million games you play. And in all the rest of the games, you'll lose the price of the ticket. Subtracting all those losses from the single win yields your net winnings for a million games, and then dividing that by a million yields the expected value for a single game.

So, if you need to decide between playing the game forever or not playing it at all, the choice is easy. If the expected value is positive, play: you'll make money. If it's negative, don't play: you'll lose money.

If you're not going to play the game forever, only a few times, then things get less clear. That's where utility comes in. But I don't like it too much. Maybe that's just because I don't understand it well enough. So I think I'll let someone else explain it. (Then, I'll complain about their explanation. )

 

[Rasmus]

So I think I'll let someone else explain it. (Then, I'll complain about their explanation. )

Okay, here it goes then:

"Utility" is a fancy way of saying "value" without meaning "money". The interesting part of mewasuring the utility of something comes in when the utility for one and the same thing changes over time.

So, if you are hungry, and I give you a hamburger, then the hamburger has a high utility to you, as it lessens your hunger.

I give you a second hamburger; you are still hngry, but it#s not quite as bad as it was before the first hamburger.

Eventually, and I am not going to guess , you will be indifferent to the next hamburger. If I give it to you, you'll eat it, but you won't mind not having another hamburger.

Even later, eating another hamburger would make you sick, so not having it would be better for you than if i gave it to you.

So, of course, utility will influence how much monetary value you place on things, too. If you're very hungry, I might get away with charging you 5 dollars for the first hamburger. That might keep you going until you find something cheaper and/or better to eat. If I sell you the first two burgers for a total of 8 dollars, you might still take two ... eventually, you wouldn't take any more hamburgers if i gave them to you for free.

Like hamburgers, money does have utility. So, going back to the question of playing the lottery:

1 single dollar - the one dollar that you have and where we are trying to decide if you should play the lottery with it - might be almost worthless to you. There is little else that a single dollar can buy you.

On the other hand, it might be the dollar that could increase the rest of your money to be enough for a pizza or the aforementioned hamburger. Then all of a sudden, spending the dollar on a lottery ticket would make a huge difference.

Also, the utility of the "next" dollar may be changing.

If I give you 10 dollars, you might be happy about that.

If i give you a million dollars you would be very happy about that.

If after the million dollars, i gave you another 10, chances are you would just look t me funny.

If I gave you a dollar every single day for the rest of your life, you might barely notice the difference. If I gave you the money upfront, it would make a huge difference - and not only because you could count your remaining days

So, the one dollar that you spend on the lottery might have a very low utility for you, that could be far outweighed by the jackpot even if you take your low chances into account. (That is the case, when a million dollars has more utility to you than the utility of 1$ times a million.) Of course, placing numbers on this is the tricky part.

I am expecting complaints ...

Rasmus.

 

[mamapajamas]

I've given up on Lotto calculations altogether.

Why? Because this actually happened at a major US lottery...

After one Lotto draw, when the jobs that calculate the prizes and winning tickets were run, it was found that the winning ticket had been cancelled and resubmitted through the system within the space of one minute.

Curious, the lottery in question called the retailer who had sold the ticket to find out what happened. The retailer remembered the incident and said that at the moment the ticket sale was run through the system, lightning struck the telephone line outside, causing a momentary power surge. He wasn't sure whether or not the transaction went through, so he cancelled the transaction and resubmitted it.

The lottery representative was startled, but tried not to telegraph that. He assured the retailer that he had done things precisely the right way, and hung up, totally blown away by what he had just heard.

Why?

What are the odds of winning the Lotto draw?

What are the odds of being struck by lightning?

What are the odds of a winning Lotto ticket being struck by lightning? At the exact moment the ticket transaction is being sent to the host?

The odds against this are cosmic in scope. It could never happen. Except that it did.

I haven't worried about odds since.

 

[rudar]

another way of framing the utility is this: what kind of life can I afford to live?

If I don't play the lotto, and I'm a not-quite-starving grad student, let's say, I might be making 24 000 a year. Rents me a dingy basement suite (OK, it's a *nice* dingy basement suite, as these things go, but still), puts food on the table, I can afford to go out for a pint every now and again, fly home to my parents every few years, drive a 30-year-old hippie van that I spend more time fixing than driving etc.

Now, if I play the lotto once a week and lose, that means I have 23 950 left for rent, food, beer, vacations. Really, in terms of those descriptions of what that money buys me, the $50 a year doesn't register. So the cost of playing the lotto in terms of quality-of-life (utility) is negligeable.

On the other hand, in the extremely unlikely event that I win, I can buy a mansion and a new truck and a sailboat and a couple of kayaks and quit one of my three jobs and have time to go play on the weekends. So the benefit there in terms of utility would be huge, even if the probability makes the average negligeable.

So yeah, in short, utility just means what benefits the money (or whatever is under discussion) gives you, in context. So while the expected value of playing the lotto is easy to calculate in cold hard numbers (it's about -50cents per game), the expected utility is harder to calculate, very much context dependent (are you playing within affordable reason, which by my definition means the lost utility is virtually nil?), but may well end up being slightly positive.

 

[BillyJoe]

POST #210

I don't think so. A very small probability is not exactly the same as a probability of zero.

I think I said "for all intents and purposes".

I'm not sure what you mean by "real chance". Would it be fair to say that your position is, "pretend that small but nonzero probabilities are exactly zero" ? That's not unreasonable. But then of course the question arises, how small is small enough.

Yes, it's so close to zero it doesn't matter.
I was talking only about playing lotto - $1 per game, $1million dollar win, odds 1:13,000,000.

Let's talk about a case where none of the probabilities is small. Suppose I want to bet on the roll of a six-sided die. If it shows one particular side---two dots, say---I'll win $4. Otherwise, I'll lose $1. Convince me it's a bad idea.

I would probably not play on principle.
That bloke is making $2 out of every six players and all he needs is a set of die?

The probability is 1/6 that I'll end up $4 ahead, and the probability is 5/6 that I'll end up $1 behind. If that's what I want to do, why shouldn't I?

I see what you're getting at.
Again I was really talking only about that lotto game, but let's have a look...
If $1 doesn't hold much value for you (you wouldn't bother to chase it if it blew away in the wind), and if $4 just comes into your radar as being an amount worth chasing then, yes, why wouldn't you?
You could reply "I'm just having a go because why shouldn't I if I want to", but then you'd be saying that only because... (see above).
If you have been considering killing yourself recently, you might even say "why not?" to a $1000 wager. There's probably lots of reasons why you say "why not?" in any situation.


POST #212

I'd guess BillyJoe wouldn't play lotto even if it had positive expectation, provided the probability of winning was sufficiently low.

I would consider each case on its merits but generally I am not interested at all in gambling. Probably because money doesn't actually hold much importance for me.


POST #214

I don't see how to give a deeper reason here than, "I just want to."

There are always reasons why you want to (practical, social, philosophical).
You might not necessarily recognise thise reasons though

BillyJoe

 

[BillyJoe]

Depends on how you word the question, I guess. My survival is by no means guaranteed if all I do is stay off of an airplane.

Am I risking my life in a more significant way then when I go to the movies - by bus?

Apparently the odds of dying on the road to the airport is higher than dying on the plane. But, of course, you'd have to add those odds in (because you have to get to the airport somehow in order to fly)

 

[BillyJoe]

I've given up on Lotto calculations altogether.

Why? Because this actually happened at a major US lottery...

After one Lotto draw, when the jobs that calculate the prizes and winning tickets were run, it was found that the winning ticket had been cancelled and resubmitted through the system within the space of one minute.

Curious, the lottery in question called the retailer who had sold the ticket to find out what happened. The retailer remembered the incident and said that at the moment the ticket sale was run through the system, lightning struck the telephone line outside, causing a momentary power surge. He wasn't sure whether or not the transaction went through, so he cancelled the transaction and resubmitted it.

The lottery representative was startled, but tried not to telegraph that. He assured the retailer that he had done things precisely the right way, and hung up, totally blown away by what he had just heard.

Why?

What are the odds of winning the Lotto draw?

What are the odds of being struck by lightning?

What are the odds of a winning Lotto ticket being struck by lightning? At the exact moment the ticket transaction is being sent to the host?

The odds against this are cosmic in scope. It could never happen. Except that it did.

I haven't worried about odds since.

It was just coincidence.
No, really, that was all it was.

Extraordinarily improbable events happen all the time.
Consider...

Question: What are the odds of being dealt 13 hearts in a game of Bridge?
Answer: 1 in 600,000,000.
Question: What are the odds of being dealt 2s, 6h, Qh, 7d, Jd, 5c, Ks, 2d, 8c, 8s, 10d, 4s, 6s?
Answer: Again, 1 in 600,000,000
In fact, those same odds apply to each and every hand in Bridge.
In other words, extraordinarily improbable events happen all the time.

The trick is to predict the event in advance.
If you, or anyone else, had predicted that event, then I too start thinking about "cosmic connections".

BillyJoe

 

[macgyver]

The trick is to predict the event in advance.
If you, or anyone else, had predicted that event, then I too start thinking about "cosmic connections".

BillyJoe

Or that your prediction was just another coincidence...

 

[BillyJoe]

Or that your prediction was just another coincidence...

Yeah but....

The probability that your prediction was just another coincidence would be so vanishingly low as to be, for all intents and purposes, zero; whereas there is absolutely no reason at all to think that the event itself was anything other than just a coincidence.


(edited to clarify)

 

[macgyver]

Yeah but....

The probability that your prediction was just another coincidence would be so vanishingly low as to be, for all intents and purposes, zero; whereas there is absolutely no reason at all to think that the event itself was anything other than just a coincidence.


(edited to clarify)

Then again, somebody will win the lotto too...

As far as the prediction goes, I guess we'd have to define the accuracy of the prediction. Are we talking Nostradamus? Or do we have time, place, names of individuals etc.?

 

[BillyJoe]

Then again, somebody will win the lotto too...

If, by a jackpot round you mean deferring the win till next week, yes, someone will win lotto with odds of 1 in 13,000,000.

As far as the prediction goes, I guess we'd have to define the accuracy of the prediction. Are we talking Nostradamus? Or do we have time, place, names of individuals etc.?

Just give me the name of next week's winner....

 

[macgyver]

If, by a jackpot round you mean deferring the win till next week, yes, someone will win lotto with odds of 1 in 13,000,000.

Just give me the name of next week's winner....

Mr. Smith

 

[mamapajamas]

It was just coincidence.
No, really, that was all it was.

Extraordinarily improbable events happen all the time.

Bingo. That's why I don't worry about odds any more. Coincidences happen, which renders the exercise of figuring odds just a game; fun, but just a game .

 

[Rasmus]

Mr. Smith

Isn't it just the funniest coincidence that I, too, thought of that name first?

 

[Rodney]

Extraordinarily improbable events happen all the time.
Consider...

Question: What are the odds of being dealt 13 hearts in a game of Bridge?
Answer: 1 in 600,000,000.

Wrong, it's 1 in 635,013,559,600.

Question: What are the odds of being dealt 2s, 6h, Qh, 7d, Jd, 5c, Ks, 2d, 8c, 8s, 10d, 4s, 6s?
Answer: Again, 1 in 600,000,000

Again, it's 1 in 635,013,559,600.

In fact, those same odds apply to each and every hand in Bridge.
In other words, extraordinarily improbable events happen all the time.

Non sequitur. If you play bridge and are dealt 13 cards, the odds are 100% that you will get one of the 635,013,559,600 possible hands.

 

[BillyJoe]

Wrong, it's 1 in 635,013,559,600.

Okay, I'll put it down to "number numbness"

Non sequitur.

Non sequitur? - show me where my logic failed!

If you play bridge and are dealt 13 cards, the odds are 100% that you will get one of the 635,013,559,600 possible hands.

....yes......so....

BillyJoe
(Rodney, I think you missed my point)

 

[BPScooter]

Billyjoe, in.re to my previous post, it wouldn't really at all change the fact that one player has equal/even/nil odds on a 6 number random lotto. But if "the more that play, the less that win" works, the big Giant Jackpot at the end of the game is shared by more folks. So please, really, anyone that has a hope from hell for a lotto: Do not send me your winning numbers. It will only halve your take-home.

 

[BillyJoe]

Heavens BPScooter, is there no hope for you, if someone sends you their winning numbers you have surely missed the boat and there will be no changing the fact of his already made gains.

Just thought you should know....

 

[Rodney]

Non sequitur? - show me where my logic failed!

You stated: "In fact, those same odds apply to each and every hand in Bridge.
In other words, extraordinarily improbable events happen all the time."

The fact that "those same odds apply to each and every hand in Bridge" has nothing to do with "extraordinarily improbable events happen[ing]." Yes, any given hand is unlikely, but it's certain that a bridge player will be dealt one of those hands. Now, if the player had dreamed the night before (s)he would receive a particular hand and then received that exact hand, that would be extraordinarily improbable.

 

[Rasmus]

You stated: "In fact, those same odds apply to each and every hand in Bridge.
In other words, extraordinarily improbable events happen all the time."

The fact that "those same odds apply to each and every hand in Bridge" has nothing to do with "extraordinarily improbable events happen[ing]." Yes, any given hand is unlikely, but it's certain that a bridge player will be dealt one of those hands. Now, if the player had dreamed the night before (s)he would receive a particular hand and then received that exact hand, that would be extraordinarily improbable.

No more than dreaming about one particular hand and then getting another particular hand.

There is no difference between me playing 14-27-33-34-40-45 in the lottery and 15-16-27-33-41-49 coming or me winning the jackpot with 2-7-16-17-30-47 being played and drawn. Both events are just as likely.

You just don't usually notice.

Yes, your chances of predicting a particular outcome are low - but it will occasionally happen, the event is rare, but in no way mysterious or special.