The lottery paradox

CuddlyHedgehog

If all possible combinations of 6 numbers out of 59 in a lottery draw have equal chance of winning, i.e one in 45 million, and the sequence of numbers 1,2,3,4,5,6 etc in our counting system is arbitrary, then it would stand to logic that an exact sequence of numbers, as we use it in our numerical system, would have the exact same chance of winning as any other irregular combination of numbers. For example the combination 5,6,7,8,9,10 should have the exact same chance as 3,9,11,45,34,12. However, empirical observation shows that the former has never happened while the latter is always the case. That means that the former pattern is statistically impossible while the latter has 100% statistical probability. This looks like an unexplained statistical preference for the selection of random numbers, as opposed to ordered numbers, which does not make sense given that the sequence of numbers 1,2,3,4,5 etc is arbitrarily chosen and should not affect the lottery draw. Any mathematicians in the room to shed some light?

Pseudonym

↪CuddlyHedgehog

The probability of getting a non-random sequence of numbers should be very small (but non-zero) whereas the probability of getting a random sequence should be very large because there are so many more random sequences than there are non-random ones.

In the sample we have (all lottery draws ever) the results are - random sequences 100%, non-random ones 0%. These are very close to the results we were expecting. 0 is very close to 'extremely small' and 100% is very close to 'extremely likely' so there's nothing odd there.

Of course, the larger our sample gets with the number of non-random sequences still at zero, the more suspicious we should be, but depending on the actual figure for non-random sequences, we might need an absolutely huge sample size before we reach that point.

You could then carry out a Chi squared test to see if the sample matched a normal distribution and if it didn't, there may be some other variable at work.

CuddlyHedgehog

↪Pseudonym

How large does the sample need to be to become suspicious? I would have thought "all lottery draws ever" should be a large enough sample?

T Clark

How large does the sample need to be to become suspicious? I would have thought "all lottery draws ever" should be a large enough sample? — CuddlyHedgehog

I would guess that a sequential lottery winning number would be pretty common. Why do you think it's not? My precise calculation, which is almost certainly wrong, is that the probability of a sequential winning number in a 5 number combination is 0.2%.

I'm not good with probabilities - hey @fdrake, help us out here.

CuddlyHedgehog

Why do you think it's not? — T Clark

Because it's never happened.

Pseudonym

↪CuddlyHedgehog

My stats is definitely a bit rusty, but at 45 million with a confidence level of 95%, I think it comes to 600. Are you sure there's not been any non-random sequences?

T Clark

Because it's never happened. — CuddlyHedgehog

How do you know? What is the basis of your statement?

https://www.lottery-guy.com/lgblog/consecutive-lottery-numbers-florida/

T Clark

My stats is definitely a bit rusty, but at 45 million with a confidence level of 95%, I think it comes to 600. Are you sure there's not been any non-random sequences? — Pseudonym

Forgot to say - my estimate was based on possible numbers between from 0 to 9. If more numbers are included, the odds decrease a lot, but they are still non-zero. See the link in my previous post.

Agustino

↪CuddlyHedgehog

Let's call sequences like: 1,2,3,4,5,6 or 12,13,14,15,16,17 or 1, 2, 4, 8, 16, 32 or 2, 4, 6, 1, 3, 5 etc. as ordered sequences. Let's call sequences like: 1, 43, 12, 23, 28, 7 as unordered. Imagine you write out all the possible results (52! results).

How many of those will be ordered sequences, and how many will be unordered?

The point is that there are many more (by the order of millions at least, but probably more) of unordered possible sequences than ordered ones. So it's no mystery that the ordered ones rarely happen.

If you had a bag with 999 white marbles and 1 red marble, and the probability of picking any one marble was 1/1000, then any white marble is as likely to be picked as the red marble. But overall, since there are many more whites than reds, whites are 999x more likely to be picked.

CuddlyHedgehog

↪Agustino

But the ordered sequences are man-made and arbitrary, there is nothing different about them to any other sequence. Only we as humans see patterns in them. We only colour them red in our mind as we are used to them being in that order. They are all white to the number selection device.

Arkady

But the ordered sequences are man-made and arbitrary, there is nothing different about them to any other sequence. — CuddlyHedgehog

But there is something different about them: they are much less likely to occur than non-ordered ones.

Any particular unordered sequence is no more likely to occur than any particular ordered one, but unordered sequences generally are more likely to occur than ordered ones generally, simply because, in the universe of possible outcomes, there are far more unordered ones than ordered ones.

This is true in just the same way that a specific sequence of coin flips like "HHHH" is just as likely as "HTHT", and yet it is more likely that one will get 2 heads and 2 tails than all heads.

Baden

Misunderstanding sufficiently clarified.

The lottery paradox

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