Lottery physics: ball placement and positions in lottery games

[shaab]

Summary:: Lottery physics- balls placement and positions

Hi all,

I am not a Ph.D. in physics or even close to that, I only had some physics classes when I was at high school .
the thing is that I like to examine things. I came across the lottery one time in my life and heard many assumptions about it as being a manipulated game (manipulated by weighting the balls or changing of air pressure and so on ..)

When I was watching the lottery one day I wondered about the machine itself being very fair and industry tested- approved by a global organization for lotteries .

I took a picture of how the lottery works in my country and checked many timings while the balls where mixed by the air pressure and picked through the vacuum,

the rules are :
the game always starts with the balls are hanging in the same order in the below vertical tubes until they release the button and they fall down in the rounded big transparent circle (nobody knows if it's vacuum inside the circle or not)

My assumption is:
If the balls start in the same position every time and fall down to the same space and the balls are equally weighted , the same order of balls should be maintained through all games when the balls are down that circle, am I wrong ? after that the air mixing time determines which ball will be picked and can't be predicted of course

What do you think ?

thank you :)

 

[anuttarasammyak]

Theoretically we can repeat the same result in tuning the machine. We have to count up all the conditions which could affect and I suppose it makes a big list. Hopefully we can design it so that every time we are successful as showed in an attempt like they did in

 

[Borek]

If the balls start in the same position every time and fall down to the same space and the balls are equally weighted , the same order of balls should be maintained through all games when the balls are down that circle, am I wrong ?

In an ideal case - yes, that makes sense. In a real case - not at all. Initial positions do vary, even if just by tenths of millimeter, release times do vary, even if just by microseconds, that's enough to produce completely different outcome each time, as the system is chaotic and very small differences are quickly amplified.

 

[collinsmark]

My assumption is:
If the balls start in the same position every time and fall down to the same space and the balls are equally weighted , the same order of balls should be maintained through all games when the balls are down that circle, am I wrong ? after that the air mixing time determines which ball will be picked and can't be predicted of course

(In addition to what @Borek has already mentioned), ideally, yes. But the system is extremely chaotic. Even tiny, tiny differences in the initial conditions will lead to vastly different outcomes. This does not just apply to the initial conditions of the balls and ball enclosure, but also the air pump, and even the air itself.

There's a whole area of mathematics to model and simulate such systems called Chaos Theory. Chaos Theory also has its own usefulness purely in the mathematical domain.

 

[physicsflea]

View attachment 294903

I'm just thinking: isn't the "true randomness" exhibited by the lottery games really an illusion? Could it be possible that the lottery balls can form a sort of "Pascal's Triangle" where predictable patterns can be deduced from it?
I don't believe that ALL lottery games are inherently "unpredictable" - though modern lottery games may have thwarted the principles of statistics and probability, there has got to be a way to forecast the next winning combination.

 

[gmax137]

there has got to be a way to forecast

This isn't a way to easy money.

 

[physicsflea]

This isn't a way to easy money.

This isn't an issue of thinking for it as "easy money" but more on challenging everyone's assumptions that the next winning combination in ALL lottery games "cannot be forecasted".
For if all forecasting models cannot predict the next winning combination, how much more with the weather, with movement of quantum particles, etc.?
I don't believe that everything in life is subject to chance - even the lottery...

 

[gmax137]

Well, consider this: has the powerball winning numbers ever repeated? Why not?

 

[Vanadium 50]

I am not a Ph.D. in physics or even close to that, I only had some physics classes when I was at high school .

on challenging everyone's assumption

Seriously, I would do this in the other order. Learn first, challenge second.

The number of bounces a ball needs to take before it's motion is dominated by quantum randomness is calculable. Its like 4 or 5.

 

[collinsmark]

Well, consider this: has the powerball winning numbers ever repeated? Why not?

This got me thinking. Allow me to provide some context assuming ideal lottery balls.

Powerball has changed its rules several times over the years. A brief Wikipedia search informs me that the current format (ball count, ball numbering, etc.) has been in place since Oct. 7, 2015, and the odds of winning the jackpot in that format is 1:292,201,338.
(source: https://en.wikipedia.org/wiki/Powerball)

With that information alone, my back-of-the-envelope calculation of the expectation value of repeated numbers (transitioning from less likely than not to more likely than not) can be calculated as follows:

First I exploited the one-to-one correspondence between a given set of winning numbers and a state of a hypothetical 292201338 state, random variable. In other words, instead of choosing a bunch of individual balls, simply roll a die (die as in "dice") with 292201338 distinct sides. Each side has a one-to-one correspondence with a possible set of winning numbers.

The probability of winning the jackpot on any given draw with a single ticket (i.e., singe set of numbers) is $\frac{1}{292\ 201 \ 338}$. (These are the jackpot odds according to the Wikipedia entry.)

If you have multiple tickets -- and ensure that each ticket has a unique set of numbers (i.e., no duplicates) -- then the probability of winning the jackpot on a single draw is $\frac{n}{292\ 201 \ 338}$, where $n$ is less than or equal to 292201338.

That means the probability of losing on a single draw is $\left( 1 - \frac{n}{292\ 201 \ 338} \right) .$

So if you start playing with one set of numbers, then on the next draw buy two tickets one with the previous set of numbers and another with a new set, then on the third draw buy three tickets, and so on; your probablity of losing every single draw is

$$P_{losing} = \prod_{n = 1}^N \left( 1 - \frac{n}{292\ 201 \ 338} \right).$$

As we increase $N$, when the probability of losing every single draw drops below 0.5, that's the N that corresponds to the expectation value of the number of draws it takes to get repeating numbers.

$$0.5 \approx \prod_{n = 1}^N \left( 1 - \frac{n}{292\ 201 \ 338} \right).$$

So now we just need to solve for $N$.

Wolfram Alpha wasn't very helpful, so I wrote a quick, dirty and ugly C# program to spit out the output.

const long odds = 292201338;
double OddsOfLosingAtAttemptN (long n)
{
    return 1.0 - ((double)n / odds);
}

double runningOdds = 1.0;
long N = 1;
while (runningOdds > 0.5)
{
    runningOdds *= OddsOfLosingAtAttemptN(N);
    Console.WriteLine("runningOdds = " + runningOdds + " N = " + N);
    N++;
}

And the result is around 20126 draws.

To date, there's only been around 1100 (approximate) draws since the current configuration was implemented back in 2015. This is far short of 20126 expectation value for number of draws for repeat numbers. So it stands to reason that there wouldn't likely be any repeated numbers yet.

If the Powerball lottery doesn't change its current configuation in the forseable future (that's a big "if"), then I would expect there to be repeated numbers starting somewhere around the year 2146, roughly*.

*(very roughly)

[Edit: corrected a typo (and copy-and-paste errors of that typo) involving the jackpot odds in the text. ]

 

[gmax137]

the odds of winning the jackpot in that format is 1:292,201,338.
...
First I exploited the one-to-one correspondence between a given set of winning numbers and a state of a hypothetical 292201338 state, random variable. In other words, instead of choosing a bunch of individual balls, simply roll a die (die as in "dice") with 292201338 distinct sides. Each side has a one-to-one correspondence with a possible set of winning numbers.

The probability of winning the jackpot on any given draw with a single ticket (i.e., singe set of numbers) is 1/292 220 338.

Not that it matters, but you switched from 292 201 338 to 292 220 338 along the way.

 

[collinsmark]

Not that it matters, but you switched from 292 201 338 to 292 220 338 along the way.

Oops. Edited and fixed.

 

[Hornbein]

In the old days they put each number in its own capsule, loaded the capsules into a cylinder, and rotated the cylinder for an hour. This failed to randomize the numbers.

The usual methods of shuffling cards also randomize imperfectly. When they went to computerized randomization bridge players could tell the difference.