posted
When I was in college, a friend of mine and I worked out this very simple alternative to statistics. We did it when I was stuck taking Intro to Stats. Basically, we decided that the probability of anything prior to t=0 is 50%, and that after t=0, it was either 0% or 100%, depending on whether it happened or not.
Okay, so that was fun in college. But I was thinking about this the other day. Every phenomenon that we've ever studied that seems simple on the surface always winds up having additional complexities under the surface. So let me throw this out as a possibility.
Suppose that when we say that the odds of rolling a 6 are 1 in 6, that's not entirely accurate. Maybe the odds of rolling a 6 average out to 1 in 6 over time. Maybe at a given point in time or space or a combination thereof, the odds of rolling a 6 are 91%. Maybe elsewhere and/or elsewhen, they're 3.5%. Maybe the probabilities that we see aren't as smooth as they appear.
It's not a scientific hypothesis, probably, at least at the moment, because there's no way to test it that I can think of. You can't roll the same die in the same way and same place and same time more than once, after all. But maybe there's some underlying mathematical virtue to the idea.
Maybe there's an underlying wave function of some kind that tends to average out to unity. If we were to find some way to detect this wave function, we could wait to roll that die until the wave was approaching its peak and thereby improve our chances of rolling a 6.
Of course, I'm actually stalling here, because I have a ton of cleaning to do today, and this is more fun.
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posted
Lisa, what is the probability that I will get all MY cleaning, kashering and shopping done, and the house ready for 6 houseguests and 18 seder guests, most of whom have frustrating dietary requirements, such as vegetarian, vegan, wheat-allergic, dairy-allergic, lactose-intolerant, and, oh yeah, "picky"?
Once I was a slave in Egypt. Now I get the same experience right here in New Jersey.
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posted
If I roll a six, how do the odds change as to my next roll also being a six?
The whole 1 in 6 rule can't be the same every time you roll. Rolling six sixes in a row have to make the odds of the fourth number being anything but a six. This might not be related to your thing, I really don't know how it interconnects, as anything involving math to me is pretty much a foreign language.
And how do you quantify chance? In poker, when I have bullets, and the other guy has cowboys, and a king comes up on the flop, the chance of my rivering an ace are very slim. But there is more to it than just odds. I've seen the unlikely happen a million times in poker, to the point where calculating the probability of what will come up next is fun and a nice guidelines, but becomes almost useless at the end of the hand. And if one unlikely thing happens, how does it change the odds that another unlikely thing will happen soon after?
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posted
Lyrhawn, that's not correct. Probability is independent of prior events, if the events don't interact in some way. Dice rolling is independent.
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posted
oddly enough, there's a scifi story dealing with something like that . . . an old del rey title, iirc. I can't recall the name, but I'll see if I can find it, perhaps.
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Here the indexing set is the real numbers, and X_0 is the random variable associated to the number 0, X_1 is the random variable associated to 1 and so on.
The notation Y = (a,b,c,d,e,f) is just meant to indicate that the probability of rolling a 1 is a, of rolling a 2 is b, and so on. which if you'll recall from intro to stat, is enough to specify a random variable.
if the indexing set happens to have extra properties (besides just being a set), you get different theories with different names, but basically one is studying stochastic processes. For instance, the indexing set could be the integers, in which case the study of what you're talking about is often called "time series". If your indexing set is the real numbers, you get what is often called a "continuous time stochastic process." If your indexing set fits more along the "elsewhere,elsewhen" usage that you mentioned, that's often called a "random field" (the indexing set here is some space,or spacetime, that you're interested in).
As you might guess, there's a well developed theory of this kind of thing, used in everything from the study of brownian motion in physics, to the pricing of options in finance, to google.
(edited to finish a correction)
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That doesn't make any sense though SwampJ. The chances of me throwing six sixes in a row can't be 1 in 6. I could throw a dice a million times and not have six sixes in a row come out.
I've played HOURS of RISK, it rarely happens.
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Lyrhawn: the chances of you throwing six sixes in a row is not one in six. The chances of you throwing a sixth six after having already thrown five sixes (or any other combination) is one in six.
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posted
I wish there was a one in thirty six chance of rolling six sixes . . . I'd clean up!
Probability of successive events isn't additive .
Think of it this way -- if you're rolling two dice, its like generating a six by six grid of possible outcomes. In only one of those thirty six spots are there two sixes. The probability of rolling two sixes in a row is one in thirty six, or one in six squared.
One in six to the sixth power is the probability of rolling six sixes in a row -- its very unlikely.
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quote:Originally posted by Tante Shvester: Lisa, what is the probability that I will get all MY cleaning, kashering and shopping done, and the house ready for 6 houseguests and 18 seder guests, most of whom have frustrating dietary requirements, such as vegetarian, vegan, wheat-allergic, dairy-allergic, lactose-intolerant, and, oh yeah, "picky"?
Once I was a slave in Egypt. Now I get the same experience right here in New Jersey.
Havah has a saying: "Dirt is not hametz, and the korban pesach is not a woman."
I had to take a break, because my back was screaming at me.
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quote:Originally posted by Lyrhawn: If I roll a six, how do the odds change as to my next roll also being a six?
The whole 1 in 6 rule can't be the same every time you roll. Rolling six sixes in a row have to make the odds of the fourth number being anything but a six.
Right. I know that standard statistics say that dice have no memory, and that may be true, but it seems counter to my experience. And it can't be that the dice themselves have a memory, so there must be some other phenomenon going on that isn't accounted for by standard stats.
quote:Originally posted by Lyrhawn: This might not be related to your thing, I really don't know how it interconnects, as anything involving math to me is pretty much a foreign language.
I used to be a math whiz until I got to college. At which point, I became a very small frog in a very, very large pond full of real math whizzes.
quote:Originally posted by Lyrhawn: And how do you quantify chance? In poker, when I have bullets, and the other guy has cowboys, and a king comes up on the flop, the chance of my rivering an ace are very slim.
Dude, you just lost me. I assume that's poker jargon, but I've never heard "river" as a verb, nor most of the rest of that stuff.
quote:Originally posted by Lyrhawn: But there is more to it than just odds. I've seen the unlikely happen a million times in poker, to the point where calculating the probability of what will come up next is fun and a nice guidelines, but becomes almost useless at the end of the hand. And if one unlikely thing happens, how does it change the odds that another unlikely thing will happen soon after?
And maybe people who have incredible luck are people who can somehow perceive these behind-the-scene functions and can tell which way the luck is flowing.
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posted
Well all stats assume a "fair die", and a "fair" roller. Especially where the latter is involved, is where you can divert from strict odds. (People don't roll the die randomly, they roll it the way their particular style is every time.)
quote:Originally posted by Lyrhawn: And how do you quantify chance? In poker, when I have bullets, and the other guy has cowboys, and a king comes up on the flop, the chance of my rivering an ace are very slim.
Dude, you just lost me. I assume that's poker jargon, but I've never heard "river" as a verb, nor most of the rest of that stuff.
Whoops, sorry, that is poker jargon. Texas Hold Em to be specific. Everyone gets two hole cards, and then the flop comes, which is three community cards, then the turn (another community card), and then the river (a fifth community card) and you make the best five card hand you can using three community cards and your hole cards. Bullets is a pair of aces for your hole cards, and cowboys is a pair of kings. Rivering an ace, means getting an ace on the river, which gives you a higher set than the guy who had trip kings.
I play a lot of poker.
quote:Right. I know that standard statistics say that dice have no memory, and that may be true, but it seems counter to my experience. And it can't be that the dice themselves have a memory, so there must be some other phenomenon going on that isn't accounted for by standard stats.
Yeah that's what I'm talking about. Stats and odds like that only work in a bubble, but even in a bubble there has to be something quantifiable in "dice memory." Is this other phenomenon at all studied and quantifiable?
quote:I used to be a math whiz until I got to college. At which point, I became a very small frog in a very, very large pond full of real math whizzes.
Heh, I was never even that good in high school. I skipped my senior year math class to take AP Biology instead, which I did infinitely better in. So I haven't taken a math class in five years. Even then I was never much good, never really cared all that much.
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posted
Dice and other basic random events have been well studied. As far as we know, there is no "probability memory" for dice, coins, or cards. This has been known since the birth of modern probability, and is relatively easy to study because a decent percentage of gamblers keep very, very good records. Your personal experience is, quite simply, a demonstration of sample bias and wishful thinking. You remember the unusual, and do not think to calculate the confidence interval for the unusual being random with the appropriate probaility over you entire experience.
For a decent and extremely readable introduction to probability and statistics, read the Cartoon Guide to Statistics (yes, it really is a comic book, and like all the cartoon guides, is very high quality).
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posted
Oh, its interesting your bring up poker, Lyrhawn. While it would be fairly simple to calculate the probability of your example (given some additional information), the possible outcomes in poker are so many that so far it is an unsolved game in game theory, much like chess (though chess is interesting in that it definitely has a solution -- given best play, either the first player will always win, the second player will always win, or it will always be a draw. We just haven't figured out which option is true, yet, much less what best play is). Poker is different because of the elements of chance and imperfect information, meaning no outcome is certain given best play.
Chess may very well be solved, or near enough to it, but it is doubtful we will ever solve poker.
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posted
I highly recommend The Cartoon Guide to Statistics for anyone interested in knowing how this stuff works, but who doesn't want to read through a text book.
The a priori odds of rolling six sixes in a sequence are:
1/6 x 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/(6**6)
much, much lower than 1/36
in fact it is 1/46656.
So, it'd be a fairly rare event in any sort of game involving successive dice rolls. One could go a lifetime and never see it.
As has been mentioned, the probability estimates assume "fair" die and fair rollers. Being systematic in ones rolling behavior is, in fact, a fair thing. Imagine setting up a machine to test whether die are fair or not. You'd want the roller to be as consistent as possible, no?
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posted
fugu beat me to it! Good recommend, by the way. Absolutely best book I've ever seen for learning basic stats/probability concepts, and a great little refresher too!
posted
Six sixes would be fairly rare, but the much more likely sequence of five sixes is something I'd expect to see every three or four games. After all, there are hundreds of rolls in a given game of risk. Six sixes would be more like every fifteen to twenty games.
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quote: What are the odds that a super-lurker would post for the very first time in this thread?
I'm actually from Ornery. Not really too much of a lurker here, because I haven't really looked at anything here very much till yesterday. Occasional glances, but that's it.
I registered a while ago. I thought I had posted before, but the stats say otherwise. In any case, I agree I haven't done much with the registration. I came on here Sunday because I was following the George Mason game on yahoo and this seemed a nice thing to look at between screen updates. Saw the topic on probability, saw what starLisa might be wanting to get at, decided to post. Maybe it was a bad idea, but oh well.
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