Lotto Probability

[Starthinker]

You'd look for a setup that resulted in the 100 heads rather than attributing the 100 heads to chance, but you wouldn't look for a setup that resulted in some random-looking sequence rather than attributing it to chance, because, even though the probability of the 100 heads is the same as the probability of the other sequence, the probability of a setup which results in 100 heads is higher than the probability of a setup which results in the other sequence.

A two-headed coin will result in 100 heads in a row. That's relatively easy to arrange. How is anyone going to prearrange for a coin to show TTTTH HTTHT HTHHT HHTTT TTTTH HTHHT HHTTH HHTTH HTHHH HTTHH HTHHH THHHT HHTTH HTTTT HHTTT THHTH TTHTH THTTH THHTT HHHTH? And why would they choose to prearrange that particular sequence rather than any other? The probability of them choosing to prearrange that sequence is basically just as low as the probability of it coming up by chance. If your explanation is just as improbable as the thing you're trying to explain, you're not really getting anywhere.

Given that you got 100 heads in a row, the most likely explanation is that someone somehow arranged that result. Given that you got the random-looking sequence, the most likely explanation is that it just happened by chance.

Once it happens, it's no use saying, "but it was improbable". It happened. Now the only question is, why did it happen?

I've been following this even though it's gotten way over my head but I'd like to jump in here. If I flip a coin 100 times and heads comes up 100 times the mere fact that you are looking for an alternate explanation indicates that it must be really, really improbable. To shorten this down lets take 20 flips. Here are three possible results:

HHHHHHHHHHHHHHHHHHHH
HHTTTHTHTHHTHTTHTHTH
HHHHHHHHHHTTTTTTTTTT

Now, statistics would say that you'd get about 50% heads and 50% tales on average. The second line demonstrates this. We would think nothing of seeing the second line. But the third line also represents this but if we were to see it, we'd think something was up. If we saw the first line we'd really think something was up.

All three lines, however, are equally probable, as far as I can see. So why do we not question the second line? Because it's after the fact. If I predicted a run of:

HHTTHTHTTHTHTHHTHTHT

And it actually came up, that would be incredibly coincidental and you'd be trying to see how I cheated, but if I simply predicted a 50-50 split and that came up, I would be accurate and no one would think twice.

So, I guess what I'm trying to say is that it's in how you DESCRIBE the probabilities and at what point in time you are viewing the results. BEFORE the flips any predictions are a billion to one, after the flips it's easy to say one combination makes more sense than another, but it's an illusion.

Not sure if my point is coming across, but it makes sense to me.

 

[BillyJoe]

Just thinking,

Well, no. Saying an event is extremely improbable is the same as saying its occurrence is rare. Six numbers in the lotto has a probability of 1 in 13 million: its occurrence is going to be very rare. A particular hand in bridge has a probability of 1 in 600 trillion: Its occurrence is going to be extremely rare! There is no way around it: improbable odds = rare occurrence.
But, as you say, one of these six numbers comes up every week in lotto. The probability here is 1.
Ergo, extremely improbable things happen all the time.
Your arguments support that statement.


Dodgy,

I wasn't asking if you would suspect a setup if you saw 100 heads coming up. I agree. As I said, I would be looking for a setup long before we got to the hundredth flip.
The question is WHY. Why do we suspect a setup when a hundred heads are flipped but not when an odd selection of heads and tails are flipped, considering that the odds of both are exactly the same?


Starthinker:

You are almost there!
You agree that the there is a seeming paradox here. Same odds but we are surprised when 100 heads come up but not when an odd selection of heads and tails come up.
You reason that if we had written down any odd selection of heads and tails and saw THEM come up we would be equally surprised. You are correct, but not quite there.
What is it exactly that causes us to be surprised when 100 heads come up?


BillyJoe

 

[Just thinking]

... If I predicted a run of:

HHTTHTHTTHTHTHHTHTHT

And it actually came up, that would be incredibly coincidental and you'd be trying to see how I cheated, but if I simply predicted a 50-50 split and that came up, I would be accurate and no one would think twice.

That's another way of saying pretty much what I said a few posts up. Each outcome is rare -- just as rare as any other -- but one outcome will result, and it will be a rare one. But not always improbable.

The other point to be made is that there are numerous ways in which a run can result in 10 heads and 10 tails, but only 2 ways in which they all come up the same. So, it not only is rare for all coins to come up the same, it is also improbable.

Being rare does not always imply being improbable -- it might, or it might not.

 

[Rasmus]

That's another way of saying pretty much what I said a few posts up. Each outcome is rare -- just as rare as any other -- but one outcome will result, and it will be a rare one. But not always improbable.

Could you please tell me an outcome for the lottery that is more "probable" than the others?

The other point to be made is that there are numerous ways in which a run can result in 10 heads and 10 tails, but only 2 ways in which they all come up the same. So, it not only is rare for all coins to come up the same, it is also improbable.

Yes, because you are now singling out one particular outcome, and measuring its likeliness to come up against a group of other outcomes.

You are saying that the lottery numbers coming out as 1-2-3-4-5-6 is special, because it is a lot less likely to come up than having one number each from the first 6 rows instead.

Being rare does not always imply being improbable -- it might, or it might not.

You haven#t shown that anyone outcome is rarer than the next.

 

[Just thinking]

Could you please tell me an outcome for the lottery that is more "probable" than the others?

There is no single outcome more likely than any other single outcome -- I have never suggested such a thing.

Yes, because you are now singling out one particular outcome, and measuring its likeliness to come up against a group of other outcomes.

You are saying that the lottery numbers coming out as 1-2-3-4-5-6 is special, because it is a lot less likely to come up than having one number each from the first 6 rows instead.

No, it is not special. It is improbable. Numbers with no special order are more likely to occur, but each one is just as rare as 1-2-3-4-5-6. But getting one (not a particular one) of the latter is not improbable.

You haven#t shown that anyone outcome is rarer than the next.

It all depends on how you describe your outcome. But no, no single outcome is more rare than 1-2-3-4-5-6. But seeing the rare 1-2-3-4-5-6 outcome is improbable while seeing a mix of numbers with no particular order is not.

 

[69dodge]

Dodgy,

I wasn't asking if you would suspect a setup if you saw 100 heads coming up. I agree. As I said, I would be looking for a setup long before we got to the hundredth flip.
The question is WHY. Why do we suspect a setup when a hundred heads are flipped but not when an odd selection of heads and tails are flipped, considering that the odds of both are exactly the same?

I said why. (But I didn't explain it well enough, I guess.) It's more likely that someone would rig things so that 100 heads come up than it is that someone would rig things so that the sequence TTTTH HTTHT HTHHT HHTTT TTTTH HTHHT HHTTH HHTTH HTHHH HTTHH HTHHH THHHT HHTTH HTTTT HHTTT THHTH TTHTH THTTH THHTT HHHTH comes up. Therefore, it's perfectly reasonable for you to suspect the first but not the second.

A bunch of coin flips result in some sequence. You are trying to determine the most likely explanation for why that particular sequence came up. One possible explanation is that the coin is fair, and the sequence just happened to come up. It is true that if the coin is fair, that sequence was unlikely to have come up. But this fact alone is not enough to reject the explanation of a fair coin. You need to find another explanation, if you can, that's more likely. It's not too unlikely that someone might rig a coin to show 100 heads. At least, it's more likely than that a fair coin would show 100 heads. But it's very, very unlikely that someone would rig a coin to show TTTTH HTTHT HTHHT HHTTT TTTTH HTHHT HHTTH HHTTH HTHHH HTTHH HTHHH THHHT HHTTH HTTTT HHTTT THHTH TTHTH THTTH THHTT HHHTH. First of all, why would they pick that particular sequence instead of any other? (It's just as unlikely for them to pick it as for it to be the random result of flipping a fair coin.) And second, even if they did pick it, how could they rig the coin to show it? So, even though it's unlikely for a fair coin to result in that particular sequence, it's even less likely that it was rigged to show that particular sequence. So, you conclude that probably the coin is fair.

(I am being somewhat imprecise here, conflating prior probabilities with posterior probabilities. But the idea is essentially right. It's all just Bayes's theorem. In calculating the posterior probability of each possible explanation for the result of the coin flips, you need to consider also the prior probability of the explanation, and not just the conditional probability of the result given the explanation. Frequentist explanations of statistical inference completely ignore prior probabilities; for that reason, they are very confusing. And confused. And should themselves be ignored. )

 

[Jaggy Bunnet]

What is it exactly that causes us to be surprised when 100 heads come up?


BillyJoe

Because we have developed an ability to recognise patterns and can see a clear pattern to this (as we would to alternate heads / tails or 50 heads followed by 50 tails), therefore we think it is a "special" result. Because we see no pattern in the vsat majority of results we do not see them as special.

 

[Rodney]

Proving negatives 101.

If you think there is something going on, tell me what it is and how to test it.

Until then, I will stay convinced that I have a fair 1:140.000.000 chance of hitting the jackpot for every line of numbers I play. It doesn't matter if I dream of the numbers, pick them randomly, use my birth date and other digits related to my past life or just play 1-2-3-4-5-6 and 0.

Should I win, it was out of sheer luck. Why should dreaming about a set of numbers be any more special than being dealt the right numbers in the quick tip game? And how would that "being special" work?

Why should I even want to look at the idea of it being special?

I don't think it's even a possibility. Should I be wrong, I'll still think I have no way of telling my special way of getting the right numbers apart from simple luck/chance/randomness. So what's the point?

There have been many verified reports of dreams seemingly foretelling the future. We recently discussed on another thread Lincoln's dream -- just a few days before his assassination -- of seeing his casket lying in state in the East Room of the White House. Regarding prophetic dreams about winning lottery numbers, I don't know that any have ever been verified, but supposedly one man won a lottery based upon numbers being revealed to him by his late son in a dream. That story was featured on "Unsolved Mysteries" and "Psychics" -- see http://www.filmscoremonthly.com/articles/2005/08_Mar---Aisle_Seat_Marvelous_March_DVDs.asp -- which I'm sure you watch religiously.

 

[Rodney]

Because we have developed an ability to recognise patterns and can see a clear pattern to this (as we would to alternate heads / tails or 50 heads followed by 50 tails), therefore we think it is a "special" result. Because we see no pattern in the vsat majority of results we do not see them as special.

Getting all heads or all tails is special in that there are only two chances in 1.27 nonillion (1.27 * 10^30) of either result occurring with 100 coin flips. On the other hand, there is about an 8% chance of obtaining 50 heads and 50 tails with 100 coin flips.

 

[Just thinking]

Because we have developed an ability to recognise patterns and can see a clear pattern to this (as we would to alternate heads / tails or 50 heads followed by 50 tails), therefore we think it is a "special" result. Because we see no pattern in the vsat majority of results we do not see them as special.

I think it is more than just being a pattern -- I think it is that patterns represent a very small percentage of outcomes (in this example) as opposed to non-pattern outcomes, and we become suspicious when an unlikely event occurs as opposed to a more likely one. Sort of like taking a random sample from a population and finding it to be 5 Z-scores away from the mean -- some red flags go up.

 

[Jaggy Bunnet]

Getting all heads or all tails is special in that there are only two chances in 1.27 nonillion (1.27 * 10^30) of either result occurring with 100 coin flips. On the other hand, there is about an 8% chance of obtaining 50 heads and 50 tails with 100 coin flips.

But there is only one way to get any combination of heads and tails, the only reason we think these are special is that we perceive a pattern in the result we get.

 

[Jaggy Bunnet]

I think it is more than just being a pattern -- I think it is that patterns represent a very small percentage of outcomes (in this example) as opposed to non-pattern outcomes, and we become suspicious when an unlikely event occurs as opposed to a more likely one. Sort of like taking a random sample from a population and finding it to be 5 Z-scores away from the mean -- some red flags go up.

All results are patterns, we just assign meaning to a small percentage of them.

 

[Just thinking]

All results are patterns, we just assign meaning to a small percentage of them.

OK, but we're talking of common every day patterns.

 

[BillyJoe]

I said why. (But I didn't explain it well enough, I guess.) It's more likely that someone would rig things so that 100 heads come up than it is that someone would rig things so that the sequence TTTTH HTTHT HTHHT HHTTT TTTTH HTHHT HHTTH HHTTH HTHHH HTTHH HTHHH THHHT HHTTH HTTTT HHTTT THHTH TTHTH THTTH THHTT HHHTH comes up. Therefore, it's perfectly reasonable for you to suspect the first but not the second

I agree with you....again!

But, consider what would happen if someone had actually guessed the outcome of the second coin flip (the random looking one). That person would be absolutely mindboggingly amazed at the sequence displayed on their screen. Everyone else, watching the flip from their own private screens, would show absolutely no reaction.
But, if the first sequence came up, EVERYBODY would be amazed.

Can you see what I'm getting at?

 

[BillyJoe]

BillyJoe said:
What is it exactly that causes us to be surprised when 100 heads come up?

Jaggy bunnet replied:
Because we have developed an ability to recognise patterns and can see a clear pattern to this (as we would to alternate heads / tails or 50 heads followed by 50 tails), therefore we think it is a "special" result. Because we see no pattern in the vast majority of results, we do not see them as special.

Congratulations Jaggy Bunnet!

Everyone sees 100 heads as being special. It doesn't need to be written down because it's already there in everyone's head. It's almost as if everyone has made a prediction that 100 heads will come up (*). If it doesn't - no reaction. If it does - absolutely mindboggingly amazing!

On the other hand a random sequence has to be written down by someone making such a prediction. And only that person is amazed if that sequence comes up, because only that person recognises that sequence as being a special pattern.

BJ


edit:
(*) I just know someone is going to misinterpret this bit.

 

[tsg]

Could you please tell me an outcome for the lottery that is more "probable" than the others?

"Someone will win" versus "I will win". It's all in how you describe the outcome. In the example of the coin flips, someone predicting "I will flip roughly 50% heads in 100 tries" is way more likely to succeed than if he gave a specific sequence of heads and tails, regardless if they are all heads or a 50-50 mix. In terms of the lightning strike claim, the chances of lightning striking the particular lottery machine as it was printing the winning ticket is very improbable, if that is the original prediction. The actual outcome, which this event qualifies as a success for, is rather "something remarkable will happen concerning the lottery" which is much more likely. In this case is it useful to look at what else would have qualified as a success. If a plane crashed into the building instead of lightning striking, would that still count? If a truck hit the telephone pole outside interrupting the power, would that still count?

It's a variation of the Texas Sharpshooter Fallacy. A shot fired from 100 yards away is very unlikely to hit one specific spot on a barn. But there are millions of those specific spots, all of which qualify as being "remarkable", making the outcome of "hitting an unlikely spot" very likely. It's no longer a question of "what are the odds he would hit that specific spot" but "what are the odds he would hit the barn at all."

 

[Starthinker]

I think discussion time is up. What numbers should I play?

 

[tsg]

I think discussion time is up. What numbers should I play?

50,52,e,pi,23 3/4, and i

 

[Just thinking]

Just thinking,

Well, no. Saying an event is extremely improbable is the same as saying its occurrence is rare. Six numbers in the lotto has a probability of 1 in 13 million: its occurrence is going to be very rare. A particular hand in bridge has a probability of 1 in 600 trillion: Its occurrence is going to be extremely rare! There is no way around it: improbable odds = rare occurrence.
But, as you say, one of these six numbers comes up every week in lotto. The probability here is 1.
Ergo, extremely improbable things happen all the time.
Your arguments support that statement.

I believe you misread what I wrote -- I will admit it can be confusing.

First off, I said "if something was extremely improbable it should also be rare in occurance." ... so you see, I agree with your opening statement.

But it does not mean that all rare events are improbable. (All Toyotas are cars but not all cars are Toyotas.) Example -- a DNA combination of a particular individual is rare -- as they are for everyone. But they (DNA codes) are not improbable, for they happen all the time -- each time someone is conceived a very rare combination occurs. But that occurance (of a rare DNA code) is not an improbable event. Expecting that code to form beforehand and predicting it exactly would be the improbable event -- not the formation of it. That doesn't happen all that often, if ever. But rare DNA combinations occur quite often.

Ask yourself which of the following you can do --- a) flip a fair coin 100 times and write down the very rare combination of heads and tales that will result, or b) predict what the rare outcome of 100 flips will be before you do it. Both a) and b) are rare events, but only one of them can be done over and over -- the other is highly improbable.

 

[BPScooter]

BillyJoe, you devil, you

Which bodies? I didn't see anybodies! And damned if I care about MC. On the other hand, I would love to know what does BP stand for?

Well, now that you ask, it has been proposed to mean "blood pressure" "batting practice" "British Petroleum" "Baden-Powell" as well as I can remember.

Real answer it is just an acronym for my given names.

Does BillyJoe mean what I think it means? You are a composer/pianist/singer? Oh, sorry, that would have an "L" in it.

OK, now I'll go read the rest of the 50 posts after this one.