# A math Problem

#### leroy

##### New Member
John is planning to play the lottery, the probability of winning the lottery is 1 in 1,000,000,000,000

tomorrow there will be an other lottery where the probability of winning is 1 in 2,000,000,000,000

the next day 1 in 3,000,000,000,000

the next day 4,000,000,000,000

after 100 days 100,000,000,000,000

the probability of winning the lottery is 1 in 1,000,000,000,000N

N = The number of days that has passed since day 1

etc

everyday the probabilities of winning the lottery are less.

John will buy 1 ticket every single day and he will play the lottery every single day for an infinite amount of time, will he ever win the lottery?

I´ll say that the answer is NO, but before justifying my answer I would like to read some opinions,

#### Rumraket

##### Active Member
leroy said:
John is planning to play the lottery, the probability of winning the lottery is 1 in 1,000,000,000,000

tomorrow there will be an other lottery where the probability of winning is 1 in 2,000,000,000,000

the next day 1 in 3,000,000,000,000

the next day 4,000,000,000,000

after 100 days 100,000,000,000,000

the probability of winning the lottery is 1 in 1,000,000,000,000N

N = The number of days that has passed since day 1

etc

everyday the probabilities of winning the lottery are less.

John will buy 1 ticket every single day and he will play the lottery every single day for an infinite amount of time, will he ever win the lottery?
Since there's a non-zero probability of winning the lottery to begin with, yes he will eventually. It might take a hundred quadrillion years, or he might win already on his very first lottery ticket. There's a 1 in a trillion chance it will happen on the first day, 1 in 2 trillions on the 2nd day and so on. All of these are non-zero probabilities, so it could happen at any time.

That's just how probabilities work, if it's above zero and you have an infinite amount of tries, then it's going to happen.
I´ll say that the answer is NO
If the answer is NO, then that logically entails that the chance of winning is 0, not 1 in a 10^12*N.

#### Rumraket

##### Active Member
This looks to me like you think something along these lines: If there's a 1 in N chance of the event, it only happens after N events.

Here's a simple way to demonstrate the falsity of this: You some times roll a 6 on your first roll of a 6-sided die.

So when you state a probability of something (1 in 100 for example), what you're really just saying is that averaged over an infinite number of iterations, there is 1 success for every 100 failures. It's not a statement about WHEN it happens. Or that successes are spaced equally with 99 failures in between every of them. That's not how it works.

To go back to your example, try to imagine he's rolling a trillion-sided die on day 1, a 2-trillion sided die on day 2, 3-trillion sided on day 3 and so on and so forth. It's now much more obvious that on any day, he can still "win".

#### hackenslash

##### New Member
leroy said:
John is planning to play the lottery, the probability of winning the lottery is 1 in 1,000,000,000,000

tomorrow there will be an other lottery where the probability of winning is 1 in 2,000,000,000,000

the next day 1 in 3,000,000,000,000

the next day 4,000,000,000,000

after 100 days 100,000,000,000,000

the probability of winning the lottery is 1 in 1,000,000,000,000N

N = The number of days that has passed since day 1

etc

everyday the probabilities of winning the lottery are less.

John will buy 1 ticket every single day and he will play the lottery every single day for an infinite amount of time, will he ever win the lottery?

I´ll say that the answer is NO, but before justifying my answer I would like to read some opinions,
You have how probabilities work all wrong. The day-to-day probability of winning the lottery doesn't change.

On each day, the probability of winning the lottery remains the same. It can be said that, if you're looking for a pre-determined sequence of events, the probability of the sequence is lower, but each individual event carries exactly the same probability. Further, as my esteemed colleague has pointed out, if the probability of winning the lottery is X, then you would expect to see a win in any random sampling of X size. This doesn't mean that it takes X attempts to win it, just that, in any given sampling of X size, you would expect it to occur once and only once. Even then, this is an average, so it can happen twice in two attempts regardless of the size of X, but then you wouldn't expect it to happen again for 2X.

More on how probabilities work HERE.

The name for the fallacy you're committing here is the Gambler's Fallacy.

In your scenario, in which a ticket is purchased every day for an infinite length of time, the number of wins will be infinite.

#### hackenslash

##### New Member
I should add,just in case you don't read the post, thatr even events with a zero probability happen all the time, and this is trivial to demonstrate.

Pick any number on the real number line. Let's say, for example, that you choose the number 7. The probability of choosing that number at random is zero. How? Because the reals are infinite, and any number divided by infinity is zero, hance the probability of choosing 7, or indeed any other number, is exactly zero, yet the probability of choosing some number is exactly one.

Naïve appraisals of how probabilities work are always going to fail, and that's even before we get into Bayes Theorem.

#### leroy

##### New Member
Rumraket said:
Since there's a non-zero probability of winning the lottery to begin with, yes he will eventually. It might take a hundred quadrillion years, or he might win already on his very first lottery ticket. There's a 1 in a trillion chance it will happen on the first day, 1 in 2 trillions on the 2nd day and so on. All of these are non-zero probabilities, so it could happen at any time.
well do the math....

the probability of winning any time after hundred quadrillion days is roughly 0.000,000,000,000,002

if you change hundred quadrillion for 10^100 days or 10^1000, or 10^100000000 you will get the roughly same probability of .000,000,000,000,002 (you need a very sensitive calculator to note the difference)

my point is that the probability converges to .000,000,000,000,002 there is a point where adding more days, wont increase the probabilities in any meaningful way.

given that the probability converges to .000,000,000,000,002 and that this is a very small number, the answer is no, john will almost certainly never win the lottery.

#### leroy

##### New Member
hackenslash said:
I should add,just in case you don't read the post, thatr even events with a zero probability happen all the time, and this is trivial to demonstrate.

Pick any number on the real number line. Let's say, for example, that you choose the number 7. The probability of choosing that number at random is zero. How? Because the reals are infinite, and any number divided by infinity is zero, hance the probability of choosing 7, or indeed any other number, is exactly zero, yet the probability of choosing some number is exactly one.

Naïve appraisals of how probabilities work are always going to fail, and that's even before we get into Bayes Theorem.

2 mistakes.

1 the fact that the series of real numbers is infinite, does not mean that I have the ability to choose from an infinite set of numbers, I can only choose from the numbers that I can imagine. I can only choose from the numbers that exist in my mind (a finite number)

2 I am more likely to imagine some numbers than others, I am more likely to choose 7 than to choose 7312004874512...

#### leroy

##### New Member
hackenslash said:
[You have how probabilities work all wrong. The day-to-day probability of winning the lottery doesn't change.

On each day, the probability of winning the lottery remains the same..
well, this is my math problem, this is my lottery and I am free to invent any hypothetical lottery that I want.............in this lottery the chances of winning today are grater than the chances of winning tomorrow.

but I do have a hidden motive, I what to know if the entropy of this universe will ever be zero. (or 0.00000000000000000000000000...1)

the probabilities of waking up tomorrow and observing a universe with zero entropy are very low ( but not zero) the day after tomorrow the probabilities will be even smaller, in 100 years even smaller, in 1,000 even smaller etc. (this is analogous to my lottery)

so after an infinite amount of days, will we ever wake up and observe a universe with zero entropy?

#### Rumraket

##### Active Member
leroy said:
Rumraket said:
Since there's a non-zero probability of winning the lottery to begin with, yes he will eventually. It might take a hundred quadrillion years, or he might win already on his very first lottery ticket. There's a 1 in a trillion chance it will happen on the first day, 1 in 2 trillions on the 2nd day and so on. All of these are non-zero probabilities, so it could happen at any time.
well do the math....
The math says it's above 0 and it can happen.

Please explain to me why a trillion sided die can't land on some particular side?
the probability of winning any time after hundred quadrillion days is roughly 0.000,000,000,000,002
So above 0, and so can happen.
if you change hundred quadrillion for 10^100 days or 10^1000, or 10^100000000 you will get the roughly same probability of .000,000,000,000,002 (you need a very sensitive calculator to note the difference)
So it's still above 0, and so can still happen.
my point is that the probability converges to .000,000,000,000,002 there is a point where adding more days, wont increase the probabilities in any meaningful way.
And it will always be above 0, so you can't logically conclude that it won't happen.
given that the probability converges to .000,000,000,000,002 and that this is a very small number, the answer is no, john will almost certainly never win the lottery.
First of all, you say ALMOST certainly, which isn't the same as never. Second, how much is "almost certainly"? Well it can only mean one thing, which is the exact probability at the particular day. So instead of "almost certainly never", it's just "John has a probability of winning on any particular day of 1 in 10^12*N". Which is above 0, so it can happen.

You can't fucking conclude from a non-zero probability, that the probability is "probably zero". What the flying fuck kind of logic is that?

What is preventing the many sided die from landing on that particular side? Nothing. So it can happen. And we know the exact probability.

#### Rumraket

##### Active Member
If you were to try to contruct an argument, using deductive logic, to derive your conclusion that it would never happen, then you would discover that you could not actually construct such an argument. The conclusion you seek can not be derived with valid logic from the premises of the thought experiment you set up.

You really need to understand what a probability is. The only way to correctly relay whether something that has an assigned probability of happening, will happen, is to state the probability. And that probability is the only correct way to answer the question "will it ever happen"?

If you answer with yes or no, instead of with the actual probability you calculated, you're making an error in logic.

It might be fine in the common vernacular to say something like practically speaking we might aswell expect events with a probability of 1 in a trillion, to not happen, but that shit won't actually fly in logic or math.

At best you could say that it's a useful approximation, or a guide, when speaking colloquially. But it's not actually true when it comes down to it. It just isn't.

#### leroy

##### New Member
Rumraket said:
So above 0, and so can happen.
there is a difference between "can happen" and "will happen"........if the probabilities are too low and you don't have enough probabilistic resources then it is fare to assume that it wont happen. (even if you are not 100% certain)

in this case, the probability of winning the lottery are very low, therefore it is fare to assume that john wont win it (even if you are not 100% certain)

And it will always be above 0, so you can't logically conclude that it won't happen.
the probability of you being hit by 100 thunders tomorrow are above cero. but given that the probabilities are too low it is fare to assume that it wont happen and you should life your life and
You can't fucking conclude from a non-zero probability, that the probability is "probably zero". What the flying fuck kind of logic is that?
I am not concluding that the probabilities are zero, I am concluding that the probabilities of winning the lottery are very low (even after any number of days) and that it is fare to assume that it wont happen........in the same way the probabilities of you being hit by 100 thunders tomorrow are very low and it is fare to assume that it wont happen

#### hackenslash

##### New Member
leroy said:
2 mistakes.

1 the fact that the series of real numbers is infinite, does not mean that I have the ability to choose from an infinite set of numbers, I can only choose from the numbers that I can imagine. I can only choose from the numbers that exist in my mind (a finite number)
Silly me for crediting you with a reasonably-functioning imagination.
2 I am more likely to imagine some numbers than others, I am more likely to choose 7 than to choose 7312004874512...
And this is irrelevant to how probabilities work.

#### hackenslash

##### New Member
leroy said:
hackenslash said:
[You have how probabilities work all wrong. The day-to-day probability of winning the lottery doesn't change.

On each day, the probability of winning the lottery remains the same..
well, this is my math problem, this is my lottery and I am free to invent any hypothetical lottery that I want.............in this lottery the chances of winning today are grater than the chances of winning tomorrow.
Well, in your hypothetical lottery in which the rules change every day, you're not going to sell any tickets.
but I do have a hidden motive, I what to know if the entropy of this universe will ever be zero. (or 0.00000000000000000000000000...1)

the probabilities of waking up tomorrow and observing a universe with zero entropy are very low ( but not zero) the day after tomorrow the probabilities will be even smaller, in 100 years even smaller, in 1,000 even smaller etc. (this is analogous to my lottery)

so after an infinite amount of days, will we ever wake up and observe a universe with zero entropy?
No, not least because entropy is something that can only ever increase in a system. The probability of waking up on any day and finding a universe with zero entropy isn't just zero, it's actually impossible, and will not ever happen.

#### leroy

##### New Member
hackenslash said:
leroy said:
2 mistakes.

1 the fact that the series of real numbers is infinite, does not mean that I have the ability to choose from an infinite set of numbers, I can only choose from the numbers that I can imagine. I can only choose from the numbers that exist in my mind (a finite number)
Silly me for crediting you with a reasonably-functioning imagination.
2 I am more likely to imagine some numbers than others, I am more likely to choose 7 than to choose 7312004874512...
And this is irrelevant to how probabilities work.
but it proves that the probabilities of choosing 7 are not zero, as you suggested earlier.

#### Rumraket

##### Active Member
leroy said:
Rumraket said:
So above 0, and so can happen.
there is a difference between "can happen" and "will happen"........if the probabilities are too low and you don't have enough probabilistic resources then it is fare to assume that it wont happen. (even if you are not 100% certain)
No, it isn't. You can't conclude "it won't happen" from "it has a low probability of happening". The last part is the only correct conclusion.

That's it, we're done.

#### hackenslash

##### New Member
leroy said:
but it proves that the probabilities of choosing 7 are not zero, as you suggested earlier.
No it doesn't, because the probability of choosing seven at random is predicated only on the available sample set, which your personal preference has no bearing on, which is why I said it's irrelevant. The only numbers that actually matter in a probability calculation are the number of available options that meet the qualifying criteria and the number of entities in the selection set. Thus, the probability of selecting any integer at random from the reals is one divided by infinity, which is zero.

You can try to play semantic games with this l[ike you do with everything else, but you can't, because this is mathematics, in which all the entities are well-defined. Sorry, but you're just wrong here.

#### Rumraket

##### Active Member
hackenslash said:
leroy said:
but it proves that the probabilities of choosing 7 are not zero, as you suggested earlier.
No it doesn't, because the probability of choosing seven at random is predicated only on the available sample set, which your personal preference has no bearing on, which is why I said it's irrelevant.
Well to be fair, it just means that when Leroy chooses a number, his choosing isn't actually random.

As we know humans are really terrible at behaving randomly (in fact they're quite predictable) and even worse at intuitively detecting randomness. We see patterns everywhere even where there are none and assign meaning and intentions to the mechanistic behavior of matter wherever we see it. Which is one of the problems with the whole debate between creationists and evolutionists, and atheists vs theists.

True, dat.

#### leroy

##### New Member
hackenslash said:
. Thus, the probability of selecting any integer at random from the reals is one divided by infinity, which is zero.

.
granted, however the event of me selecting a number is not a randon event and it does not impply infinite opptions.
I should add,just in case you don't read the post, thatr even events with a zero probability happen all the time, and this is trivial to demonstrate.

well then please provide your demonstration, ................so far all you did was to provide a non random event that does not involve infinite options. .

events with a zero probability happen all the time
that sounds to me logically incoherent, but I´ll give you the opportunity to prove your statement..........provide an example of an event with zero probability that can happen ...